I've FINALLY worked it out.
The headphones that I was using are Sony MH1C that I use with my phone that have an inline mic.
On Windows when the ASIO driver is not installed, they work on the JA11, even having a mic input.
But I get that "No valid sample rate" error when the ASIO driver is installed. It works with other headphones that have no mic though.
I discovered that I can use the Fiio Control panel on my phone to disable the mic, and then it works on the PC as well!!
What an ordeal!! Would have been nice to have some kind of documentation...
Anyway after all that, I can wholeheartedly recommend the dongle.
It fits everything that I wanted. Small, good latency with ASIO driver when I'm using a DAW, and can do simultaneous output from Windows audio at the same time as using the ASIO driver.
I also tested the dongle with my Beyer DT880 Pro (250 Ohm) and it definitely could drive them way too loud for my liking, so power wise that's pretty impressive for a tiny dongle.
So yeah, really great value, especially because I got my JM12 for $3 US...
But I will probably buy a Fiio JA11 later on to support them. I wanted to test if it'll work for me before laying down the big $$$, lol.