Your question may have more success at r/redsox.

The Fenway public tour is good, especially on a non-game/event day. You get to see more on those days. On a game/event day, parts can be closed off.

I never tried a birthday tour and the site says nothing, so I cannot help with that.

If you like tours, you may want to try the New England Sports Museum at TD Garden. There's lots of sports memorabilia spread out around the Garden. It's a good tour if you know local sports well.