Induction motors, if from a good manufacturer, are inherently long lived – giving many thousands of hours operation as long as routine maintenance is carried out. They normally only fail because of the incorrect operation by its users.
Bearing failure or over-loading are the chief culprits, unless operating in decidedly damp conditions (and that includes starting, without drying out, after long term storage). If your motor is stopping by disconnection of an overload trip, you need to find why – or it will eventually fail (as it seems that it is not the overload trip that is set too low on this occasion).
Running the motor completely off load for a period may (but not always) give you a clue as to whether it is an internal motor problem, or the external mechanical load that is the problem.
Single phase motors without any protective overloads are far more likely to be destroyed than those fitted with proper starters with appropriate safety devices included.
Motors are expensive but protective devices are cheap. The “Buy cheap, buy twice” adage comes to mind.
Any decent 3 phase inverter will likely trip if the phase currents are not in reasonable balance, I would think. Three phase motors are not that difficult to find, but the extra cost of a VFD needs careful consideration – as there are other benefits, for motor protection and speed control, built into them. There are non-variable single to three phase converters out there too.
Two of my machines have well documented vibration reports, when driven by single phase motors, which makes me lean toward 3 phase motors wherever/whenever possible and appropriate.
My advice would be to sort out the problem of the current motor. Not just fit a larger powered motor (if that is the case). Otherwise you may finish up with two blown motors or a damaged machine.