Rats with the Lumberjack upgrade will occasionally stop working unless you, push them or interact with their menu.
This tends to happen when you're away, less so when you're nearby but still happens under supervision (Still within the chunk or neighboring chunk, but not right near him.)
For reference he also has the replanter upgrade on him.
And it is also in a LAN environment.

Minecraft Version: 1.20.1
Rats Version: 8.0.6

I can't really get this to happen. Doesn't matter if you rejoin the world or unload and reload the chunk, the rat keeps working just fine. I noticed sometimes it gets stuck trying to collect a sapling off a block it can't reach, although this seemed really rare.
Starting in 8.1, rats wont fire their pathing tasks as often, allowing them to reassess things a bit better. Once 8.1 comes out (hopefully in a few days) let me know if this is still an issue and I'll continue to look into it.