In my experience, your request is unrealistic. Bluetooth shouldn’t be too much of a problem, either built-in or with a dongle, but a stick that can emulate much of anything from the PS2 library is going to be ridiculously expensive compared to other options. I’m not aware of any Fire Stick, however new, that could do it. The majority of sticks are good for up to 16-bit, and quite a few that are at the upper end of the sane price range will stretch to PS1, some N64 and Saturn, and maybe even some Dreamcast and PSP.

If you have to have PS2, I’d really recommend going for a micro-pc or high-end Android box if it absolutely has to be as small as possible.