I saw a Kling for sale about 10 years ago for $30k was a “brand new” 3C table in a men’s club in NorCal somewhere. It had sat there since new and wasn’t really played on for 80? Years at the time. Was so nice wow!
With this table in the OP, I’d do all the homework possible and talk to the smartest guys in the wood refinishing biz and restore it to what ever was original to a Kling in its period. If it’s gloss-then gloss, if it’s something-then figure out what the something else is. Butchering a historical table like this is not good. I’d just make sure you are doing the correct process and using the correct materials. Then full blast-fun restoration project.
Getting the table historically accurate would be my goal.
Enjoy the process and congrats. Wonderful table, they play good too. I’ve played on many of them over the years.
Best
Fatboy

One of my all time favorite tables