I am also facing a similar scenario these days with my Liana’s AC. I have done some research on internet and I think that I have found the solution.
Problem: Once the compressor starts then it protests with some sound which is normally heard once you have loose belts.
Diagnosis by Car Mechanic: Initially I thought that it was a problem with loose belt which was tightened by my mechanic but he says that a problem is coming from compressors bearings. This was verified from the car mechanic.
Reason for the Problem: Normally these sounds are the first for most sign that your compressor is about to fail and the reason behind is the absence of oil within the cooling system.
False Statements by AC mechanic: I have visited several AC mechanic but the problem is
all of them tries to RIP you off with false statements. They know practically how to clean various parts and how to recharge the system but most of them are so DAM illiterate that at times while trying to RIP you off they screw your entire system.
Diagnosis: Previously Summers once I took the Car for AC servicing then I remember that Ac guy told me that he will have to take the older gas out in order to completely service the car and would charge me Rs 2500 for this Job. I agreed, he must have taken the old gas out and would never have bothered to put AC lubrication in the system while charging it again.
So right now cooling is good but compressor protests with strange sound.