I had a look at Toolforge's documentation.
Indeed, that seems like a nice place to host the bot. We can schedule pretty much everything we want, and jobs are queued in the event of a higher load on Wikimedia Cloud Services.
I'll just need to setup myself a Wikimedia Developer account. Since I don't want my current email adress to go public, it'll take a few days until I can get a "custom" mail address that would be dedicated to that purpose. Either way, we have plenty of time ahead of us.
The bot is now installed on Toolforge. https://toolsadmin.wikimedia.org/tools/id/lingua-libre-bot
There's also a "deployment" changelog here: https://wikitech.wikimedia.org/wiki/Nova_Resource:Tools.lingua-libre-bot/SAL
While the bot is installed on Toolforge and already due to execute every day at 2AM, it will not be "working" properly over the next few days, since I need some more practice to get used to Toolforge and Kubernetes. It's therefore running in "dryrun mode", which means that while it actually queries stuff from Lili and computes other stuff, it won't apply the changes on the wikis.
I'm considering this issue as resolved. Lilibot will rise for its ashes any time soon.