Essentially, I want players who join a game to (be more likely to) join servers where the round is close to completion, or in between rounds, to avoid excessive waiting times. Most decent round-based games have this in place, yet I cannot seem to find any documentation on customising the roblox “Server Fill” algorithm. If anyone can let me know if this is possible and if so where I can find documentation I would greatly appreciate it.
Thanks 
As far as I know, the best way you can implement this is to teleport the Player as soon as they join into a server where a round is about to start. You can try looking into using MessagingService and send data about a server to all servers, whenever the round status changes (e.g. if round finishes, sends a message to all servers; if a round just started, send a message to all servers etc.)
I might try this, my only issue with the redirection is that wouldn’t the immediate switch to another server feel kind of clunky (since the player is essentially joining twice)? Especially when compared to games like arsenal that (seem to) immediately place you in a server with a fresh round. I might play around with it to see how much of a delay immediate redirects actually create but I’m really curious as to whether it’s possible to influence join placement directly.
I’m pretty sure it’s the only way. Also, try to use teleport data to prevent triple teleports.
Bricklabs does singleplayer this way and it isn’t too bad. In fact, I find it kinda convenient.