29 June would be another non-weekday, much like Year Day.

That works for me!

Which months would you (re)move to adjust Sep-Dec to 7-10?

I would get rid of July and August because I don’t care for the Caesars. I guess we could expand the format and have a Quinquember and Sexamber.

Personally, I would just rename all of them but no idea what they should be.

Sounds like a fun idea to do with friends and keep it going as a running joke!