Yes the power surge of the TV could cause a UPS with a depleted battery to power sag causing the DVR to reboot. Most UPS will run a self check when they are powered on check the status of the indicator lights on the UPS front panel.
For now I would disconnect the TV from the UPS.
What size and how old is the UPS most need to have the battery replaced every 2 years it may be cheaper to replace the entire unit.
Note: TiVo's do work very well with UPS my DVR's, Computers, cable amp, phone and Network are powered through good quality UPS's that have AVR. I designed the system to survive 20 to 40 minutes without power giving me enough time to power down devices.
VicW, Ex-Comcast customer ... And loving it!.