Most of the VFDs I dealt with in at least the last decade have had an input for a motor thermistor.
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Besoeker, most high end VFDs do now come with a thermistor input, or have an analog input that can be configured as a thermistor input, that's true. Unfortunately what is also true is that here in the US (and probably Canada), most standard motors are sold with no embedded thermistor or other type of thermal device. They are generally special order so nobody waits that long. Inverter Rated motors tend to come with thermistors, but if he doesn't have a VFD on there now, chances are that this is not an Inverter Rated motor.
Other issues raised by ice worm:
You usually can accelerate a motor more often using a VFD, as long as you can live with a long acceleration cycle. Might be with considering here, in fact the longer acceleration time may preclude the need to start and stop so often, but its hard to tell because we dont really know the nature of the process. But that may also mean changing the motor eventually.
Attaching a thermal sensor to the outer case may be problematic in that there can be a 30deg C difference between the outer case temperature and the winding temperature, so without at least some empirical data from which to factor it, the information is not all that useful, except as a way to determine eminent failure. Kind of like the "Temp" light on your car; by the time it goes off, it's already too late.
Re: increasing the fuse size and letting the OL trip. I agree that is likely what will happen if the fuse size in increased. But also I think that although this would give a better indication as to what is really going on, in my experience it tends to set up a new form of failure. The operators will immediately now complain that the overloads are too sensitive and either demand that they be turned up, or do it themselves, moving the problem right down to the motor windings and rapid failure of the motor.
So I too think this really is a process change issue. Determine whether or not that motor was de-rated for the higher starts per hour and if it was, put in a way to limit the minimum time between starts and the maximum starts per hour that this is rated for. If it was not de-rated, well, good luck with that, better have a spare motor handy...