I assume that you tried it and it didn't work. You can't upgrade HDMI. You might try connecting the Firestick to the TV and change the video resolution to be compatible with the receiver. That might limit the video quality so the solution would be to leave it connected directly to the TV and get your audio from the TV to the receiver with HDMI-ARC or optical audio.
Indeed I tried it and it didn't work. There are HDMI 2.0-to-1.4 converters, but no to-1.1 converters, as it seems. My whole home cinema system including a large CRT projector is about 10 years old, and has no HDMI ports, except for the HDMI 1.1 ports on my AVR. I tried an HDMI-to-SCART converter, which worked but the quality was just too poor on my 95" wide projection screen. Next I'm going to test an HDMI-to-YPbPr component converter to see whether this provides an acceptable solution for the Firestick.
But I really wanted to avoid a digital-to-analog conversion, hence my question.
If your projector has RGBHV (not the same as YPbPr) or VGA that should work better than scart.
I've used graphics and data grade CRT projectors in the past and suggest It might be time to consider updating the projector. It won't take a lot of money to get a projector you should be happy with. Probably better much resolution and an enormous increase in light output even if you have 9" guns.
I tested the HDMI-to-Component+R/L Audio Converter along with my existing Cinemateq Picture Optimizer on my CRT projector and 95" wide screen, and the results are very acceptable. I'm relieved I don't have to squeeze out the money for new home cinema equipment for now.