Is it possible to power a solar MPPT charge controller from a backup DC power supply?
In principle you can do this.
In practice there may be some "issues".

You will probably need to rectify the AC and certainly should do so even if the controller handles it as it will be a non optimum input.  An MPPT controller MAY have input rectification but this is a non essential part of the basic design and if present would probably be for protection against reversed polarity input. Input should be reasonably well smoother DC - not just rectified AC. High ripple AC may well interact with the MPPT algorithm which would be quite likely to "chase its tail" trying to keep up with the AC variations - which are MUCH faster than the usual input variations. Even if the sensing input is filtered enough to not track the AC ripple it could still cause problems as it may reach different conclusions about the RMS value of the input from what the inverter proper sees.

Best would be to smooth the rectified AC - the UPS may have some capacitors that you could use for this.