I'm a noob, please let me know if this is the wrong place for this kind of post. I bought the cheapest 1080p projector I could find on Amazon, and while it works about as well as I expected I'm experiencing some difficulty pairing it with my Fire Stick remote.

My goal is finding out the 4-digit number that will tell my Fire Stick what IR codes to expect so that my Fire Stick remote can control my Jimveo projector.

The Fire Stick has a menu for putting in an IR profile number, and I've tried a smattering of profile numbers I've found online with no success.

So I whipped up a little photodiode circuit, probed it with my oscilloscope, and pointed my Jimveo remote at it. I have discovered the following:
1. The remote uses 38kHz NEC protocol.
2. The remote sends "0x00ffa857" (LSB-first, so literally "0b0000 0000 1111 1111 0001 0101 1110 1010") for ON/OFF, followed by repeat bits every 108ms.
3. The remote sends "0x00ff8877" (literally "0b0000 0000 1111 1111 0001 0001 1110 1110") for MUTE, followed by etc.

I could fill out the rest of the code table if necessary, but so far I've been able to rule out every code table I've been able to find using just those two. Is there some way to determine or at least narrow down the candidates for IR profile based on a code table?

I'm happy to provide more IR bitstreams, if that'd be helpful. My cobbled-together setup can do COMMAND -> HEX BITSTREAM lookup, but not the reverse.

The buttons on the Jimveo remote are as follows: ON/OFF, MUTE, REWIND, PLAY/PAUSE, FAST-FORWARD, UP, DOWN, LEFT, RIGHT, OK/ENTER, BACK, MENU, SOURCE-SELECT, VOLUME-, VOLUME+. There are no serial numbers (that I can find without full disassembly) on the remote.