2011 Honda CR-V AC Issues

I've had my CR-V since December of 2017, I'm due to pay it off by the end
of this year but the AC went up about a month ago. It suddenly started
blowing hot air on one of the hottest days of the year. I took it to my
mechanic and they said it was the compressor so they replaced it for about
$1100.

When I picked it up it was blowing cold air for the first minute or two but the
AC kept going in and out on the 10 minute drive home. I thought maybe it
just needed to run a little longer but the next day it wasn't blowing cold air at
all. It would blow cold for the first few minutes if I was running it early in the
morning when it wasn't super hot out but it would always start blowing hot
air again and if it was already hot out it wouldn't be cool at any point.

I took it back to the shop and they said when the compressor when up it
probably took out something in the climate control panel (or something like
that) and they ordered a new one. They just replaced it and when they were
testing it they said it worked for about 10 minutes but then the AC stopped
working again and now they think the compressor going up messes up
something in the computer. They said the computer part alone is about
$850.

I generally trust my mechanic but I'm worried they are just going to keep
trying things that cost a lot of money that don't wind up working. I don't
know if the compressor really caused these issues or if the compressor was
never the problem in the first place and they are just trying to cover up the
fact they charged me $1100 for nothing.

Has anyone else had this issue? Does any of what the mechanic said sound
plausible? Should I take it to the dealer to try to get a second
opinion/diagnosis or should I just take it to another mechanic?