Normal scripting tools get you far; .sh in unix, maybe Powershell if you are a Windows guy.
I have done this with .sh scripts and once their capabilities run out, custom C programs, but that's only because I like to work with C. Many would choose Python scripting instead.
In any case, if you are capable of writing the AVR firmware in C, some production automation code (in C or whatever) should be easy compared to that.
Basically, get some SCPI power supplies and multimeters etc. Like
echo "OUTP ON" > /dev/ttyUSB0
Then just call avrdude in your script. You can loop it until it returns success, this way simply detecting when the AVR is connected to the fixture.