If post#2 is correct and the headphone is 10 Ohms, this is more like a loudspeaker than a headphone. Even the headphone output is probably not going to be all that effective. Use a small and inexpensive amplifier with adjustable gain AFTER the Ultralite mk5 and connect the headphone to the speaker outputs using an adapter. You will be able to get plenty of drive voltage into that load, but make sure that you can start at a low volume and adjust as necessary.
Good idea, but then I hope that the TDH39 wiring is using the seemingly common (but not universal) two separate mono 1/4" plugs, which would mean a lot more choice of potential candidates for speaker power amps. If the cable only sports a stereo 1/4" plug instead, the amplifier would need to have a traditional single-ended output with both L and R output grounds connected together. Many Class D amplifiers in particular have bridged (BTL) outputs with both + and - terminals being actively driven, which means L- and R- must not be shorted together under any circumstances. That includes a lot of small and inexpensive models. The two plug wiring can accommodate these just fine, the stereo jack cannot. (Side note, a model with PFFB would be recommended for better frequency response accuracy, although with a frequency range of up to 8 kHz only it wouldn't be a truly major concern.)
Also, ideally you want a model of modest gain, like around or sub-20 dB. Even then, there should still be enough maximum output to let the magic smoke escape from the headset. Output noise level should be quite low as well. Even relatively small drivers strapped to your ears are much more sensitive than loudspeakers at a distance of not much less than one up to several meters. From that perspective your best bet may be a 3e Audio A5 (gain 16 dB with volume bypassed, SINAD 105 dB at 5W/4 ohm) or perhaps Topping PA5 II (25 dB, same SINAD), both $200 and change and thus the cheapest anywhere near on the chart by far.
Now obviously, the variant with a stereo jack could be accomodated with a decently stout headphone amplifier instead, as previously discussed. That should generally make noise a non-issue, although driving 10 ohms may be a bit of a stretch as loads below 16 ohms aren't commonly considered. I would still trust an Atom Amp+ / 2, Topping L30 II (L50) or similar to do a decent enough job.
BTW, an Ultralite in this application is like buying a schoolbus for hauling your groceries. It can be done but is neither fast nor cheap, and many seats will remain unoccupied.
Now if the headset wiring is of the two plug type, I can at least understand where the idea came from. The mistake was going by mechanical compatibility while not realizing that things were a poor match on the electrical side (the expected load for one of the line level outputs would be in the 600 ohm to 20 kOhm range, with very little in terms of power transfer). In this case, going for a set of mechanical adapters (bare wire or possibly banana plug on twin lead speaker or power wire to 1/4" mono jack, tip = +) means that you can easily use an amplifier designed to drive 4-8 ohm loads to much higher power than you'll ever need. That's actually to the point where you need to watch out for problems on the opposite end instead, i.e. noise.
