Just to be clear, when the Belkin UPS unit is on AC power, I presume there is zero voltage out the DC plug?
Since the original pictures are not showing anymore, I made a new diagram of how I think this works...
This is my presumption: if I hook it up as in the diagram below, when there's normal AC power, the controller will just see the 24V from the original power supply, but in a power outage, the Belkin UPS takes over and supplies 12V to the controller (the AQLink T1 controller apparently switches to a battery mode). What I don't want is for both the Belkin and the original power supply feeding 12+24 Volts into the controller at any time! Is this correct? I hope I can do this without needing diodes, etc.