Low operating current, but high stall current. How to pick a driver chip?
This is a common issue for large AC power systems.  The starting current of a large induction motor can be quite large.  So large the NEC has a bunch of special rules to allow larger breakers, thinner wire, and the like, where that would not normally be permitted in any other case.  

One of the solutions in the industrial setting is to use an electronic controller to limit startup current - for instance a flux vector drive does this as well as allow fine speed control.  Of course that is for AC induction motors. 

Methods differ for controlling other motor types, but the same concept could apply.  For instance an induction motor will give you plenty of back EMF if you control it properly... a series-wound DC motor won't, and you might just have to have a DC-DC converter be part of your controller which is capable of the startup current it needs at the least voltage it can live with, which would be quite low.  

This can be part of a "prevent the motor from overheating" strategy, e.g. on a DC motor limiting current to continuous-safe-for-commutator so it can start moving the moment the user frees it; but also occasionally spiking current to see if the motor can get unstuck on its own.