I don´t think the E10K is suitable for HPs with 250ohms.
It´s Okay for anything up to 80ohms max I think.
Not enough power has it´s ways into the sound quality of course.
Problem is the power....those integrated things usually don´t have beefy HP amps.
So I´d sugguest TWO devices.
1. SMSL M100
2. Sabaj pha2
alternative dac is a
S.M.S.L Sanskrit 10th. (77€)
Can´t think something decent cheaper than that.
Maybe lucky shot on a Khadas board ?
I´ts a bit more....but way up in quality.
The higher price is easily justified with the huge quality boost.
Of course this setup has more power for headphones too. I´ve got the Pha2 here and tested with my DT1990s.
Get´s very loud and sounds pretty good but of course my main setup with SMSL SP200 is another story which is damn fine considering the price
I had time and was able to get the Pha2 for 40€ for example But even the usual 60€ price is really justified. The M100 is and was always around 75€.
Then there is the SMSL M6. If you can get one for around 100€...good device too. Not a powerhouse though. But remembering it...it was able to drive my DT1990 fairly well. Pha2 is clearly stronger though.
Topping DX3pro is of course pretty much the best you could think of...but I understand that is out of your budget.
/EDIT: Then I found this one....but don´t even ask me anything about that thing...Information given is really good looking at the price.
https://www.amazon.de/FX-Audio-Verstärker-Bluetooth-Koaxialer-Schwarz/dp/B08BHP4JXD/ref=sr_1_3?dchild=1&keywords=FX+Audio&qid=1594602674&s=ce-de&search-type=ss&sr=1-3