For what it's worth: I have two Texas Instruments 8000 stations. They are unique among the various stations I have, in the fact that they have a 12 v Gel Cell 7 amphour battery inside the metal box. The charger is not the ordinary wall wart, but a wall wart that has been made especially for keeping gel cells topped off. The batteries were changed when I got them, and have been perfect for over 5 years, so they are probably getting close to needing a check with the battery charge/discharge device I got from a Radio Controlled Airplane hobby store. But the message is that you just don't hook a regular wall power supply to these.
In any event, they have a very simple wiring that the station pulls power from the battery which is kept charged. When the grid power fails for whatever reason, the battery keeps the station and it's recording alive. The displays are still showing wind, rain rate, etc.
To be frank, our power has been remarkably reliable. Brief outages with lightning strikes at the substation and between here and there, but usually it trips and is back on line within seconds. This means resetting every darned clock EXCEPT these stations, and with my computers on UPS for filtering and brief outages, I still get my history of the storm.
You say your station takes 5 volts, so a 6 volt battery MAY be OK without over volting the station, or you could put in a little voltage control chip and just generate a little heat to make it furnish 5 volts.
Gel Cells aren't cheap any more, and 6 volts (other than the sporting stores for fish finders and such) are hard to find, especially in larger amp sizes, but that is what eBay and Amazon are for, right? Or if price is no barrier, go to Batteries 'R Us or something. It just depends on what you want to do. I find some places with computer stuff sell UPS solutions as a loss leader and are far more versatile than a pure battery back up.
Dale