I have an iPod Touch and it won’t drive your Aeolus. I’ve been using it to burn in my Verite overnight and even on full volume, it’s at comfortable listening levels. You’d need a portable DAC/amp to drive “difficult” headphones, with the added benefit that the sound would also be better. I use a now-discontinued Oppo DAC/amp and it’s night and day, even with easy to drive headphones like the Focal Stellia. So I’m guessing iPod Touch is off your list because you’d need 2 devices connected by multiple Apple-approved cables/adapters.
If you scroll to the bottom of this post by @antroid , you’ll see a really useful table of DAPs that he’s compared. Or if you want to live dangerously, read that post in full, because you may realize that the Sony NW-ZX507 is what you need. It’s more than your budget but maybe there will be a good Black Friday price.