After a full week of trying to fix this and a few days of running with multiple patches, it seems as though there are still problems with federation. So I'm resorting to crowd sourcing: if you notice any problems interacting with other instances (voting, commenting, posting), or you're from another one and can't interact with FMHY, make a comment here. Include two links: one on the side without any issues, and one with them (if possible.)
Example post:
New communities may not federate properly.
This will hopefully pinpoint what instances and/or types of activities are causing problems and why - are they using CF or not, are their servers just blocking us, is it specific to posts made on FMHY or does it include FMHY posts made on other instance's communities, stuff like that.
As for current workarounds:
There is a userscript known as "LULS" (Lemmy URL Link Switcher) which mainly redirects all links (on other lemmy instances) to a configurable host instance. This script seems to do some funky stuff with federation, and sometimes manages to properly sync posts or communities when used. (Comments may not work after a post has synced.) If you're worried about safety, edit it to only allow usage on the domains you want.
Try the regular federation process first. Some instances seem to behave differently to this (mostly lemmy.world during my tests), but searching does occasionally work properly.
I'd also, again, like to ask for anyone with technical knowledge on Lemmy. I'm now currently the only staff member on this and even though we don't have many people here anymore, it doesn't hurt to try or ask.
I'd also like to thank anyone for sticking around for this long, particularly u/Martineski and u/mr_MADAFAKA. Also thank you to dbzer0 for his help on this problem so far. Without everyone still here, we would've stopped running this instance, so thank you :)