Is there a Way to Force Google Photos to Upload in the Background?

No. There is no way to force Google Photos to upload in the background i.e because of iOS limitations. Google Photos app for iOS stop working in background in few minutes. But, simply make sure that the Refresh feature is enabled for Google Photos app on iOS device. Also, enable the auto-sync feature when you are on Wi-Fi network and your iPhone is plugged in.

Quick Tip to ensure your Photos never go missing
Photos are precious memories and all of us never want to ever lose them to hard disk crashes or missing drives. PicBackMan is the easiest and simplest way to keep your photos safely backed up in one or more online accounts. Simply download PicBackMan (it's free!), register your account, connect to your online store and tell PicBackMan where your photos are - PicBackMan does the rest, automatically. It bulk uploads all photos and keeps looking for new ones and uploads those too. You don't have to ever touch it.