There are three possible culprits:
1: Partially clogged AC system (Condenser, Expansion Valve or one of the hoses itself)
2: Overcharged AC system
3: Weak engine compression
With the passage of time, tiny metal dust particles from the wear and tear of compressor accumulate in the condenser lining making it harder and harder for the compressor to blow pressurized liquid freon through it. Eventually a time comes when either your AC system leaks out on you in the middle of a hot day due to O-rings blowing apart, or the compressor itself fails. Better to replace the AC condenser at least every 100K miles instead of waiting for a disaster.
Overcharged AC system is also a reason why the compressor can "take load" as desi mechanics define it. When there is too much freon in the system, compressor is on the verge of locking out trying to compress already compressed liquid freon which could potentially destroy the compressor itself or the clutch at a bare minimum. Freon quantity should NEVER exceed the official factory specs. EVER!
Last possibility is an engine with worn out rings or a leaked head gasket. When an additional load engages with the crank shaft, the engine bogs down and the RPM gets low. A healthy engine with a healthy AC system will never get bogged down.
My car's rpm stays at a good healthy 6-800 with or without the AC at all times.
Solution: Get a set of AC manifold gauges and test your AC system for over-charging which is unfortunately very very frequent in our country due to illiterate mechanics and so-called ac technicians everywhere. Consult the service manual for details. Here's a snapshot for reference.