I'm curious if anyone here has suggestions on how to train and improve on climbing once a week.

I can climb once a week

I have access to a weight room M-W-F
I own a hangboard, the Metolius Simulator

Could anyone suggest a training plan for me?

This has been covered a lot, but here you go....

Check out "The making of a Rock Prodigy" by the Anderson brothers. Check out Steve Bechtel's site ClimbStrong. Figure out what will work for you as far as weight training (Steve's site) and hangboarding (Rock Prodigy) then work on climbing hard routes/problems with good technique the one day you do have available.

Hope this helps.

For someone that started climbing in late July this year, you don't need training. You just need a lot of climbing and other workout that you do. Just be sure to do enough workout on the opposing muscles so you don't get injured while advancing too quickly.

Depends on your goals, but there's a good article here.:

How to improve your climbing: Pro rock climbing coaching for mortals