How well do microcontrollers deal with unstable power supplies?
In theory they should be fine since you aren't violating any specs.  However, there is probably a upper frequency limit on supply voltage variations that will cause trouble.  You can only guess what that might be since datasheets don't tell you.

Do what you can to limit the frequency of the fluctuations.  You always need a good high frequency bypass cap on every power lead to the nearest ground lead.  If your supply voltage might otherwise fluctuate at high frequency, add additional capacitance.  For example, a 1 &micro;F ceramic for every power lead, and then maybe 10 &micro;F or more near the chip where all the power leads are tied together.

Consider adding a ferrite chip inductor in series with the supply before all the capacitors mentioned above.  This should be followed by 10s of &micro;F capacitance, plus the individual bypass caps.  The inductance and capacitance form a second order low pass filter that attenuate high frequencies significantly.

The drawback is that there will be some voltage drop across the ferrite chip inductor.  These are designed for this kind of use, so have only a few 100 m&Omega; at most, but that can still be a problem in some cases.  It helps to keep the heavy current draw off the microcontroller power feed.  If the micro needs to drive LEDs, for example, have the micro control transistors, which then control the LED current coming from some other supply or from before the chip inductor.

Ultimately, you'll have to talk to the manufacturer to get a real answer, if you're ever going to get one at all.  The best strategy is to avoid being in this situation in the first place.