Will a non-dimmable LED work if the dimmer switch is on full?
Yes, LED bulbs typically work fine on dimmers set to max.  In my experience, even the "non-dimmable" ones don't seem to be harmed when dimmed although your mileage here may vary with bulb and dimmer models.  

This seems to be a big improvement in comparison with "non-dimmable" CFLs, which seem to be very susceptible to destruction - I have destroyed several after just seconds of dimming.