Since the room exists and is reasonably accessible i figure the relay can either send something from the room to somewhere else, or from somewhere else to the room. Can explain "bypassing" the room by doing a somewhere->room, room->target double teleport, but that seems like inefficient usage of the energy hungry relay.

For fast traveling out I haven't changed anything. It's up to you to take the elevator back up if you want to "save energy", or do it from anywhere inside if you need to get out fast and don't mind the extra energy use of a double teleport. 

Again, this is a really simple mod and it doesn't actually keep track of energy usage or anything of the sort. All it does is move the fast travel destination. Have tested it by doing a full institute play through with this mod active and didn't run into any issues. 

Combines well with Institute Teleportation for Survival Mode for immersive survival mode fast-travel. When you arrive in the institute there's a delay as the script in that mod re-activates all the fast travel destinations, but just imagine it as the relay taking some time to recharge. Only takes like 5 seconds, depends on the destination.

XBone version: https://bethesda.net/en/mods/fallout4/mod-detail/4063196
PS4 version: https://bethesda.net/en/mods/fallout4/mod-detail/4063199
The Institute Teleportation for Survival Mode mod does not have any console versions and I'm not the author for that mod, so not much I can do about that.