No! This is not true. For some reason the config was only recognized on the second attempt and now your script works. Sorry for the confusion and many thanks for your help!
So all my debugging was done on the wrong IF. But it teached me a lot..
For reference:
Wed Jan 15 14:51:59 2025 daemon.notice netifd: eth1 (4189): +CIREPI: 1
Wed Jan 15 14:51:59 2025 daemon.notice netifd: eth1 (4189): +CNEMIU: 0
Wed Jan 15 14:51:59 2025 daemon.notice netifd: eth1 (4189): +CEREG: 1,"AD99","00A25202",7,0,0
Wed Jan 15 14:51:59 2025 daemon.notice netifd: eth1 (4189): offline -> registered - home network, TAC:44441 eNodeB:41554-2
Wed Jan 15 14:51:59 2025 daemon.notice netifd: eth1 (4189): +EONSNWNAME: 1, 0, "1&1", 0, ""
Wed Jan 15 14:52:00 2025 daemon.notice netifd: eth1 (4189): AT+COPS=3,0;+COPS?;+COPS=3,2;+COPS?
Wed Jan 15 14:52:00 2025 daemon.notice netifd: eth1 (4189): +COPS:0,0,"1&1",7
Wed Jan 15 14:52:00 2025 daemon.notice netifd: eth1 (4189): +COPS:0,2,"26202",7
Wed Jan 15 14:52:00 2025 daemon.notice netifd: eth1 (4189): Registered to 1&1 PLMN:26202 on LTE
Wed Jan 15 14:52:00 2025 daemon.notice netifd: eth1 (4189): Activate session
Wed Jan 15 14:52:00 2025 daemon.notice netifd: eth1 (4189): OK
Wed Jan 15 14:52:01 2025 daemon.notice netifd: eth1 (4189): AT+CGACT=1,1
Wed Jan 15 14:52:01 2025 daemon.notice netifd: eth1 (4189): +CGEV: ME PDN ACT 1
Wed Jan 15 14:52:01 2025 daemon.notice netifd: eth1 (4189): OK
Wed Jan 15 14:52:02 2025 daemon.notice netifd: eth1 (4189): AT+CGPADDR=1
Wed Jan 15 14:52:02 2025 daemon.notice netifd: eth1 (4189): +CGPADDR: 1,"10.111.190.136","0.0.0.0.0.0.0.0.0.0.0.0.0.0.0.2"
Wed Jan 15 14:52:02 2025 daemon.notice netifd: eth1 (4189): OK
Wed Jan 15 14:52:03 2025 daemon.notice netifd: eth1 (4189): AT+CGCONTRDP=1
Wed Jan 15 14:52:03 2025 daemon.notice netifd: eth1 (4189): +CGCONTRDP: 1,5,"internet.mnc023.mcc262.gprs","","","61.8.132.52","8.8.8.8","","",0,,1500,,,,,,,,,,,,0
Wed Jan 15 14:52:03 2025 daemon.notice netifd: eth1 (4189): +CGCONTRDP: 1,5,"internet.mnc023.mcc262.gprs","","","42.0.15.188.34.0.169.153.1.16.54.52.0.2.0.1","32.1.72.96.72.96.0.0.0.0.0.0.0.0.136.136","","",0,,1500,,,,,,,,,,,,0
Wed Jan 15 14:52:03 2025 daemon.notice netifd: Interface 'eth1' is now up
Wed Jan 15 14:52:03 2025 daemon.notice netifd: Network device 'eth1' link is up