You don't describe what you were doing that caused you to be pulled over, so it is hard to say exactly why the citation was warranted. However, driving with multiple significantly under-inflated or flat tires poses a danger to others on the road, so even though it is understandable that you were driving that way because of the tires, it probably was indeed negligent. The expectation is that if you can't install spares to get you to the mechanic, you will have the vehicle towed, so as to not pose a risk to others. Driving negligently does earn you a citation under ch. 90 sect 24(2)(a).
Given that you had to stop driving the vehicle at that point, you should, in principle, have been able to call AAA yourself to have them tow the vehicle the rest of the way. Police typically are only authorized to order towing through a specific service (whoever holds the municipal contract), so if the officer called a towing service on your behalf, there probably was no choice of carrier. By law, there are limits on what can be charged for towing and storage, and the municipality can also set a lower maximum rate. These seem to be the current rates. The regulation also says that if the destination is other than the carrier's garage, the rates may not apply. The rates are based on distance, service / waiting time and labor, so it is not necessarily possible to compute the cost in advance.
If the operator said that the charge would probably be $120 and it ended up being $140, that seems like a reasonably close estimate. On the other hand, if he promised you that it would cost $120 and then the minute the officer left he revised the figure up, that smells of crookedness. There is almost certainly an official means of filing a complaint with your municipality.