Probably yes... I've used a whirlpool start device kit that includes that brown relay on multiple brands. You're not likely to do any damage to the compressor if it doesn't work. Just ID which terminals on the relay are currently used and try that configuration first.
I've seen plenty of rigged compressors that I was surprised were working. Assuming your compressor is the standard 3 pin style you should be able to come up with a combo that works. Just be certain neutral goes through the OL or PTC and then to common pin.

I know a little bit about electricity and appliance repair through DIY experience, but am still a self-proclaimed rookie looking to fix my wine fridge. I am looking to replace the same relay discussed here in this post. My relay is definitely the problem. I tested it and can see burn marks, etc. I bought the same replacement from Amazon that you said worked for you, Electric Woot. How did you get the third whole (pin?) hooked up since the replacement only has 2 pins? That third pin is at the end of the wire that connects with a plastic piece that won't fit on the new replacement relay.