Car heating after turning AC on could be due to excessive load on engine. But that may not be the cause altogether and over heating could be due to already poor condition of cooling system.
Check condition of radiator, fan speed, hoses and water body. Service radiator, add a high speed fan (if required) and replace water body if corroded, add coolant and T-stat valve.
When cooling system is all set work on AC. The compressor has probably been cooked throwing soot into the condensor and dryer bottle. Sadly you will have to replace all these components. And will cost round about the same amount you ve been quoted. Kabli compressor is used most of the times known as duet compressor. It works ok but be sure about its condition. Condensor and dryer bottle should be bought brand new.
If you are unsure about the diagnosis visit a few mechanics and take their opinion as well before replacing the AC parts.