Beenthere is correct, what the hvac company did is correct.
Min. Ckt. Ampacity Amperage Minimum: 29.3 - means the running amp on compressor or the wire size for the condensor. #10 wire is rated for 30amp which is higher than 29.3, it's good. The compressor has its own thermal overload protection.
Max Fuse or Ckt. Bkr. Fusible/Coupe Circuit (HACR per NEC): 50 - means the starting amp on compressor. maximum breaker can be 50 amp, 40 amp breaker is ok as long as it does not go over 50 amp, and beenthere is right if the breaker keeps tripping, they do allow a little higher as long as it isn't larger than 225% of the circuit rating or compresser rating, whichever is larger.
Everything your HVAC Did is corect even if they were to keep the 30amp breaker as long as it doesn't trip the breaker. Can you use a #8? Absolutely, but not neccessary according NEC.
Do us a favor and tell your electrician to worry about his trade and not someone elses. If he still argue print this link out and give it to him.
sorry, not being an ass to the electrician, But I had a same similary problem with a handyman that caused me to wasted my time going back and forth.