Most state codes prohibit central AC for garages, for the reason you mention.

You will need to withdraw around 200 cfm from the central system, if you can insulate the ceiling.
The house needs to intake outside air make up for that non-returned volume.

If your air handling has a fresh air intake, there is no problem.

If not (99% of residential AC have none), you will not get 200 to feed the garage.
Doors can leak in as much as 10 cfm.
Toilets' exhaust fans and drier vent can leak in more when not working.
Each toilet exhaust fan will exhaust around 50 cfm, while working against your garage.

Results:
There would be less AC for the house.
The area will be conditioned even if you are not using it, but you will pay for the used electricity (unless you install a damper to block the garage supply at will).
Humid air infiltrating the house, which will be constantly negative, thru doors and walls (from the attic via power outlets).
Fumes from the garage (from glue oddors to car exhaust) will infiltrate the house.
Mold inside walls will develop with time.

Any mini-split is more expensive than any window shaker.
I would insulate the ceiling, seal the garage door, and install a 9,000 BTUH window unit in the window (or 12,000 BTUH if garage door or window is facing South and big water heater is inside the garage).

For better air distribution and less noise inside the garage, I would install a mini-split of similar capacity.