The GF has a power filter built-in which protects it from power surges. A surge strip and most battery backups have a power filter built-in. When you daisy-chain surge strips, the power filters can work against each other, causing you to have the sort of degraded power that you’re trying to avoid and possibly cause one filter to burn up or start a fire.
About 10 years ago at work, I personally witnessed my office mate’s surge strip catch on fire at the outlet the 2nd surge strip was plugged into. It was mostly a loud pop, with a bright flash of light from that one outlet, then smoke rolling out. He unplugged it from the wall and I grabbed the extinguisher. The plug from the 2nd surge strip was welded into the plug of the now melted first surge strip.
Nearly 20 percent of all failures and fires stemming from surge protectors happen as a result of daisy chains, according to a publication by the U.S. Office of Compliance. https://www.prairielectric.com/blog/can-i-plug-a-surge-protector-into-another-surge-protector/
If you don’t want to believe this particular article, Here is what APC has to say: www.apc.com/us/en/faqs/FA158852/ There are countless others to find if you search “can I plug a surge strip into a surge strip?”
An APC or battery backup can be OK as long as it does not have surge protection on the outlet you plug your GF into.