Thank you very much bmork, this was the hints that made the issue move forward! And I do apologise for my late response, especially as you put your valuable time writing down the essentials.
Now I'm at the point where I can send AT commands and get real responses.
For future referense, some information to summarise for other having the same issues:
This post discusses the AT comands for a similar module: Mikrotik R11e-LTE-US with Openwrt - #2 by silverchris.
Use RNDIS USB Dongle for WAN connection:
Basically, plug in the module and run the following commands on SSH:
opkg update
opkg install kmod-usb-net-rndis kmod-usb-acm kmod-usb-storage
After this there shall be rndis interface installed called eth1 (or usb0).
Make WAN interface. Follow the instructions here: https://openwrt.org/docs/guide-user/network/wan/smartphone.usb.tethering