Thanks to excellent suggestions from Cahudson42 and Beershaun, you now have your separate and low-cost music player, which should give you decent sound quality (SQ), while streaming AmazonMusicHD. Though the fact that LG does not list the audio DAC in the specs sheet probably means that it is not on par with their higher-priced phones. In the future, if you replaced the LG Rebel 4 with an LG V10 or higher, or LG G7 or G8, you would get quad ES9218P DAC-Amps and pretty good SQ.
If you wanted still better SQ, you could simply plug into the LG Rebel's USB port a high-quality dongle form factor DAC-Amp such as the Tempotec Sonata HD Pro ($40) or E1DA #9038S ($105, but it takes only a 2.5mm balanced cable plug) or the upcoming E1DA #9038D (takes a standard 3.5mm three-pole plug). The LG Rebel would then be used only as a "transport" to send your music file or stream to the dongle. The SQ of this combination would be extremely difficult to achieve with a DAP at any price. The E1DA #9038S would provide plenty of power to drive even higher-impedance headphones. It would take a high-quality desktop DAC and headphone amp to rival and perhaps surpass the Rebel+dongle combo. The arrival of high-quality USB dongles seems to render DAPs obsolete. The only (small) downside of the combo is that you have a full-fledged phone plus a dongle, perhaps a little inconvenient when you are out and about, especially if you also have to carry your iPhone XR with you. For people that do not use a separate device as a music player, using a dongle on their phone will also drain the battery some.
The compact HiBy R3 Pro that I bought last December for $200, which replaced a HiBy R3 that I used for a year in my car, has noticeably better SQ than the R3. It has a THD spec of 0.0008% (which is -102dB) using its balanced 2.5mm socket and driving a real load, which is quite respectable for a DAP at that price. It also has dual DACs compared with the R3's single DAC, and delivers twice the power as the R3 out of its balanced socket (210mW per channel into 32ohms; however only 21mW into 300ohms, which is weak for high-impedance headphones). I cannot understand how HiBy continues to sell the R3 at the same price of $200. The R3 Pro comes with either dual CS43131 or dual ES9218P DAC-amps (the Saber variant), and its SQ is comparable to what you would get directly from a LG V10, V20 or from Qudelix 5K, FiiO BTR5 and Shanling UP4. Anyway, all of this about the HiBy R3 is moot because it does not fit your use case. It runs a HiBy OS, which is a custom stripped down Android to enable decent UI response on its weak Ingenic CPU. It also provides stripped down Tidal and Qobuz apps for streaming over WiFi but no offline mode, and most importantly no AmazonMusicHD streaming app currently (or AppleMusic either). HiBy says that more streaming services are coming, but I am skeptical about that. I use it only to play music files from a microSD card when it is inconvenient to carry a full-sized DAP somewhere.
To get close enough to the SQ of the Rebel+dongle combo for practical purposes, while not having to deal with a dongle, and while being able to stream any music service you like, you would need the iBasso DX160, which at $400 lies outside your stated price range. In the DAP world, the DX160 is a bargain for what it provides. Its specs and SQ surpass even those of iBasso's much higher-priced flagship (but older) DAP, the DX220 with the default Amp 1 Mk II. It uses dual CS43198 DACs, and through its balanced 4.4mm Pentaconn socket (it also has a standard 3.5mm socket) it achieves a SNR of 130dB and a THD to signal ratio of 0.00022% (-113dB), per iBasso's Audio Precision measurements. Through the balanced output, the DX160 can generate up to 6.4Vrms to drive 136mW into a 300ohm load, which will make even most high-impedance headphones plenty loud, and which is likely more than most phones are capable of even with a powerful dongle. I have the 2019 version, but the 2020 version beefs up the output buffer to provide even more current for 32ohms lower impedance headphones. The DX160 runs Android 8.1, which allows you to get any streaming music service over WiFi. Furthermore, the audio layer of the OS is customized to bypass AndroidSRC so that all apps playback audio in bit-perfect mode, and HiRes audio is sent to the CS43198 DACs without downsampling to 16/44.1. Of course, you would need to check online forums beforehand for user experiences to verify that the AmazonMusicHD app streams properly on the DX160, and that its offline mode works. I have not yet streamed any music on my DAPs; my streaming service usage is limited to my desktop PC.
Now that you have your independent source-dac-amp, the biggest step up in SQ would come from replacing your headphones. Try parametric EQ on your current ones first, to see if you can get a preferred tonality that would suffice. Finding better headphones at a reasonable cost is a perilous journey, and solderdude's website that Cahudson42 linked to is an excellent resource to start your research. Ultimately, measurements and reviews can guide, but not guarantee finding a headphone that matches your ears and taste.
The Bluetooth (BT) you use is the weak low-SQ link in your setup. I do not hanker after HiRes audio (16/44.1 is good enough for me, I think it improbable that any difference in SQ between that and HiRes would be audible to me), but I would say that even the best Bluetooth codecs available now are lossy and may be just about able to match 16/44.1 resolution at best. Sending HiRes audio through BT is pointless. The default SBC has a low default bitrate; AAC is better but you need an Apple device to transmit the BT. If you need the convenience of BT, to get decent SQ you should aim for at least AptX HD codec support in both transmitter and receiver. For this, the transmitter must have the Qualcomm CSR8675 chip, and the receiver needs either the CSR8675 or one of the newer lower-power receive-only Qualcomm 5100 series chips. If both transmitter and receiver support even higher bitrate codecs such as LDAC or UAT, you can get very good SQ over BT.
If you are going to get better headphones in the future, I recommend you use a lightweight BT receiver such as (all using dual ES9218P) Qudelix 5K, FiiO BTR5 or Shanling UP4 (or even perhaps the Hidizs AP80 Pro DAP also with dual ES9218P DACs, though its listed THD+N is a high -96dB) that you can clip to your shirt pocket or belt, rather than buying headphones with the BT built-in. The latter locks you in to a particular BT implementation and a particular set of headphones, and some of the weight of a BT receiver still goes into the headphone earcup. A independent BT receiver gives you the flexibility to use any headphones or earphones depending on your mood, and can be changed out for a better BT implementation when one comes along. The upcoming BT 6.0 standard will supposedly come with better than SBC and AAC codecs built-in by default, that may perform as well as the current best proprietary ones. I use two reliable but now dated EarStudio ES100 BT receivers which give me AptX HD reception. While I use one, I recharge the other so that I effectively get infinite battery runtime, which would not be possible with a single more expensive headphone with the BT built-in. I think the HiBy R3 Pro and iBasso DX160 are capable of receiving (and transmitting) BT using AptX HD, LDAC and (R3 Pro only) UAT. They are also capable of receiving regular WiFi from your iPhone, LG Rebel, laptop/desktop computer or presumably the internet via your router, and with the WiFi your music streaming would be lossless (but consume the battery faster than BT).
You can get excellent BT quality, approaching the CD 16/44.1 information rate (losslessly compressed bitrate), with your current gear by adding to it as follows. Get a USB OTG cable (should be inexpensive) and the tiny Shanling M0 DAP, and use it with your LG Rebel to transmit via LDAC codec BT. See the post
Bluetooth aptX HD from Windows 10 PC to Topping D50s by Berwhale and his subsequent posts in that thread. It will require a firmware update of the Shanling M0 to enable LDAC transmission capability. The FiiO M5 might also be capable of this function (it can transmit LDAC BT), but it is not known to us, whereas Berwhale has verified that the M0 can do it. Buy one of the Qudelix 5K, FiiO BTR5 or Shanling UP4, which can each receive LDAC BT 5.0, clip it to your shirt and plug your Sennheiser BT HP into it with the existing cable. Or better yet, buy a balanced cable for the HP and plug it into the 2.5mm balanced output of the BT receiver. Voila, when the devices are BT paired, you have a high bitrate LDAC BT stream. Then, when you are of a mind to sit down and seriously listen to 16/44.1 or HiRes music, you unplug the Shanling M0, plug a E1DA #9038S or #9038D into the LG Rebel via USB C, unplug your Sennheiser HP from the BT receiver and plug it into the dongle for the highest SQ listening.
On a side note, the fact that your Sennheiser HP has a cable with a 2.5mm four-pole plug going into the HP means that your HP can likely take a balanced cable that you may find on Amazon or AliExpress (2.5mm four-pole at one end and either 2.5mm four-pole or 4.4mm Pentaconn or even 4-pin XLR at the other).
