Is it ridiculous? yes. Do you have any recourse with the contractor or home warranty company? Not really, except to stay on top of the promised repair schedule. When my compressor went down due to end of life failure, they had to order a new one from the supplier which takes a solid week to be fulfilled and ship. The replacement compressors for older models are not one size fits all and seldom used, so they usually need to be ordered on demand from the supplier. I think you're lucky that your warranty company actually found a vendor who properly diagnosed the issue and was authorized to replace the compressor. That's usually a $1000-2000 repair. The bad news in my case is that a year after we replaced the compressor on my 20 year old heat pump unit, it finally started to leak elseware and the heating cooling capacity diminished and we determined it was time to replace. so hopefully your compressor buys you some more time, but know that other issues can still develop.
That timeframe is frustrating but relatively reasonable. It's your responsibility to provide alternate window AC units or portable AC units for tenant to use in the meantime until the repair is complete. My PM has portable AC units that I rent from them and they deliver to the property until the repair was completed. That's a landlord responsibility and you need to handle that ASAP. A gift card really isnt a reasonable substitute. Also, as an aside, most warranty companies are extremely slow to approve and make repairs using a 3rd party contractor (who gets paid a low rate to handle your claim and prioritizes their non-warranty repair calls). By choosing that route, you have to accept that repairs would not be done quickly and goes with the territory.
One last idea: look on craigslist for 1-2 portable AC units or window units, buy them used, provide them to your tenants, then sell them when no longer needed to get back most of your cash. or keep them in your storage for next time AC fails at another rental.