Question is .... what should I look at and where to start?
That issue is known, looks like it's a combination of the weak built-in antenna of the RPi's and probably some bug in the connman or IWD codes. Log files probably should help, but with problematic network isn't easy to make it.
The rock solid solution is to use LAN cable if you can. If not, the second is to use a Wifi-Lan adapter, no needs any driver, it just "translate" the Wifi to LAN cable net. If you have around a Wifi-USB dongle, with a "real size" antenna, you could try it, the question is if the LE have the right driver for it... And if you are already using LE 12.2 version, should try the latest 12.2 nightly. Have included the latest updates.
I have an RPi3B with Wifi connection, modded for external antenna (for this need a very good soldering skill!), latest LE 12.2 nightly, and working fine.