Have you tried this setup:

In the office: Eero router LAN port (Ethernet cable) -> (Ethernet cable) goCax adapter 1 (coax) -> (coax) office coax port ->

In the bedroom: bedroom coax port (coax) -> (coax) goCax adapter 2 (Ethernet) -> (Ethernet ) bedroom switch

Do you have any “blank” wall plates/cover plates in your house? On the walls or in the ceiling? If yes, check under those.

Per another comment, a tone generator might help. The “wireless”/probe end of the generator will work through walls of the cables are close to the drywall. It’s still a guessing game, but it might give you some clues.

Post some pics of the RJ11 terminations (take off the wall plate). As long as there are four twisted pairs, there’s a good chance you’ll get gigabit speeds over them.

Can you contact the builder? They might have a low voltage plan and/or remember the approx locations.