From your description, the A/C contactor is unlikely to be bad.

AGC automotive fuses are fast acting. For electromagnetic loads such as contactors or motors, fast acting fuses are usually the incorrect type and not acceptable.

A time delay or "slow-blow" is probably required. Look at the VA rating on your transformer and divide by 24. (i.e. if it is 75VA then the maximum current would be 75/24 = 3.125 A). Install a fuse sized at or just below the transformer rating that is the slow-blow variety (the type with the heavy or spiral filament). A low voltage rating is not necessary. A line voltage rating such as 125v or 250v is perfectly fine.

Edit:

I concede that the answer to this problem depends on some specifics that are lacking.

If the OP fuse is a glass cartridge style, then it is likely that a slow blow (MDL type or similar) is required in place of the fast acting automotive type. If the fuse is the automotive blade type as Mazura pointed out, then fast acting is the only option for those and the problem lies elsewhere.

Electrical contactors do draw relatively high current until the contacts seat into the closed position, so I also have to agree that a sticky A/C contactor could blow the fuse even if the contactor coil is actually good. To test if the contactor is sticking, I would turn off the outdoor disconnect to the compressor and open the condenser access panel for inspection. Turn the thermostat to "cool" and watch the contactor pull in. If the contactor action is sluggish, that could cause the problem. With power to the compressor turned off at the disconnect, this test could be performed repeatedly.