Leaking/Perished Service Valve Air Con

If you paid them for moving the unit not for servicing the answer is: No!

If they were more customer oriented they could have had a quick look or offered the service as an add on.

You also could have asked for them to service the unit while they are at it.

Whatever was agreed between the two of you it seems they did not have to service the unit.

I'm assuming this is a mini-split system, where you have an outdoor condensing unit, and an indoor evaporator air handler.

You indicate a service valve, that from your description was on the indoor unit. I can't think of any unit I've seen with a service valve on the indoor unit. Also these typically use flare fittings as brazed joints are a fire hazard to installation where this mini split systems are installed.

As you called the same incompetent person to fix their 'handy work' any evidence of what the true failure was is now lost.

Do you have photos of what was leaking?

Also is this just a single evaporator system, where you only have this one air handler inside the living space?

My guess is that an improper flare joint was leaking, or the 'technician' broke this unit's flare fitting by over torquing during the move of the indoor unit. And the leak was damage to the internal tubing connected to this flare fitting.

It's likely that to have the unit moved they sent an installation mechanic....Now to troubleshoot and repair the unit you have a service mechanic...You should have had the unit started and tested before and after the move....In any case it still would cost to have the unit repaired...the service valves are not cheap, and after installation requires filter drier, evacuation and full recharge...It would have been cheaper to replace the valves during the move, because the move requires the same procedure of filter installed , evacuation, and recharge....Some people seem to think anytime someone works on their A/C unit it is now somehow magically guaranteed to not fail, the unit isn't guaranteed, just the work that was done....

does it look something like this or a valve on a bicycle tire?

If you had ANY inkling as to the actual cost of moving it, you simply should have left it where it was and put that money towards a new one for your bedroom.

If you had no inkling, then you did not get the job priced/agreed before agreeing to it......if you did, how much more was it? Only pay up to original price + 15% max....

Old units rarely like being moved.....like old humans.....

Either way, bad move!!! (Pun intended!)