In the Developer options (made visible by tapping on the Build number in Settings - About phone a few times until you are being declared a developer) there should be a switch “Mobile data always active” … perhaps toggling this into the state it is not in now might make a difference.
(If it is enabled now, disabling it might trigger mobile data being turned on successfully when leaving the Wi-Fi network. If it is disabled now, enabling it might keep a working connection in place.)
Else …
Regarding OS install from scratch, the Android 13 install file wasn’t released yet … https://support.fairphone.com/hc/en-us/articles/360048050332