I’ve never tried this with a telecoil receiver, but in my experience, anything that can output audio via a headphone jack will stream from the Roger On if properly connected. However, I’m not sure that approach makes sense if you plan to use it while walking around museums or on public transit. To use the headphone jack, the Roger device has to be inserted into the base unit, and it’s pretty unwieldy to carry it around like that. Plus, it needs to be connected to a power source. You could probably rig something up with a portable charger brick, but then you’ve got several clunky devices precariously daisy-chained together, and they’ll all be knocking against each other in your bag. I’d be worried about damaging the USB port on the Roger On, because that arrangement is just begging for it to get bent somehow. Personally, I’d switch to a battery aid before I’d mess around with that.
Maybe one of the professionals can chime in here: Can a 15-year-old with OP’s audiogram use Phonak’s Audeo Lumity RT? That one is rechargeable and has a telecoil.