The problem seems to be the failed Zip signature verification.
Since I use LineageOS and more current TWRP versions from twrp.me, I can’t pinpoint what specifically is going on here.
“Zip signature verification failed!” is known to happen if you try to install something other than one of Fairphone’s OSes with their TWRP 3.1.1-0.
Solution would be to install a more current community TWRP … Fairphone 2 … but you are not trying to install something other than one of Fairphone’s OSes.
“Zip signature verification failed!” is more trivially caused by “Zip signature verification” being enabled in TWRP’s Settings (you can get there from the main menu).
But I just booted Fairphone’s TWRP 3.1.1-0, and this setting doesn’t seem to be enabled by default.
In my understanding TWRP can only verify Zip signatures of signed Zips it has the proper signature key to.
As I would suppose Fairphone’s update Zips would be properly signed, and as I would suppose Fairphone’s own TWRP would have Fairphone’s signature keys … I don’t know what exactly fails here.
If it was me, I would boot a signed, more current TWRP image without flashing it to the phone … see twrpwoflashing … and try to install the update file with that … “Install” in TWRP’s main menu, then navigate to /cache by tapping “(Up A Level)” and “cache”, and hope fp_update_gms_whatever.zip is still there to select.
But then again I’m a tinker kid with a backup of my data, and perhaps contactsupport is a better idea, they certainly know their own stuff better than me 
.