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Get a Raspberry Pi 5 or a mini PC instead of a stick, or something FPGA-based like a MiSTer if you don't mind spending some money.
Pi 5 isn't too expensive. Raspberry Pi 4 was already pretty good for emulation, and the Pi 5 is noticeably faster than the Pi 4.
Firestick is way too underpowered to emulate PS2 games. At best you're going to get N64 and PS1 with inconsistent frame rates. It'd work fine with NES, SNES, Sega Genesis etc though.
Alternatively, maybe ask if on the go gaming is their thing. I have played more games since I got my rg35xx sp than any steam sale before (there are many others of similar features at similar price points, more expensive ones to get PS2 and GameCube capable emulation). Waiting for someone? Game. Appointment call taking long? Game. Work lunch time and have some minutes to spare? Game.
Game changer. In my case I sacrificed capabilities (ps1 and n64 only for easy to emulate tittles), but I can even play GT2 on the go, what I hoped GT for PSP should have been. Depending on when they grew up, it might be a worthy sacrifice for portability.
That's what I suggested.
He then sent me THIS LINK and said he was looking at this.
He's dead set on a stick format. I look at those controllers, and assume they play like shit. The crazy thing is, unlike me, he actually knows how to use and navigate linux.
I have a raspberry pi 4, that I want to like, but I can't figure linux out to turn the damn fan on. So mine sits, mostly unused. Whereas he would know what he's doing. He's a software developer by trade.
But he wants a stick. So I'm hoping someone knows what to do.
Let him get a stick then and laugh when his PS2 games run at 2fps. Simply isn't possible yet in stick format. Rpi4 won't even do PS2 in playable framerates
Rpi4 won't even do PS2 in playable framerates
I don't have a Pi 5 to try (I've just got a Pi 4), but apparently PS2 is doable, at least for some games: https://www.reddit.com/r/RetroPie/comments/1bekm56/playstation_2_emulation_on_raspberry_pi_5/kv261pp/ (sorry for Reddit link)
So...get him this stick. It even says in the product pictures: Perfect Gift For All Ages I suspect your brother-in-law is of any age so it fits.
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.