Short answer: no, not really. Longer answer (with practical advice at the end):
The problem here is that Unit('K K-1') compares equal with Unit('1'). Quickplot is detecting this and not bothering to label them at all. We could in principle change this logic so that it sees that Unit('K K-1') is not the same as Unit('1') and proceeds. If we do this we also need to redefine how the units string is chose. Currently we use the unit's symbol when it is shorter than the string form of the unit (except for time), which for your case:
>>> Unit('K K-1').symbol
'1'
>>> str(Unit('K K-1')
'K K-1'
you'd still just get '1' and not the units you wanted, because '1' is the shortest representation. Re-writing all this logic is certainly possible (not too much code), however it is essentially a subjective choice. One scheme of logic is unlikely to work for all common cases, and we must accept we cannot please everyone. The current scheme does OK for most things, even handling things like Unit('g kg-1'), but your use case is one that is not handled in the way you expect.
Given this my suggestion would be to avoid quickplot and just use iris.plot. Quickplot is nice when it does what you want, but if it doesn't then fall back to iris.plot instead. All quickplot does is figure out the names of the axes and the name and units of the quantity plotted, then add labels and a colorbar. You can just do this manually, and if your cubes have some commonality e.g. same axes this maybe very simple.