I haven't used that product,but there is another powdered bleach/cleaner out there that will work if used correctly.
https://cheapcues.com/q-clean-shaft-cleaner/
It's been a LONGGGG time since I used it because I found a better method before the Magic Eraser came along,then switched to it because it's faster than the old method,and perhaps even a tad more effective if used to it's full capability.
However,using ANY of those methods will 100% change the way the shaft feels when you are done,so be prepared to deal with restoring it to your own liking if you are doing it yourself,and understand that if you are having it cleaned by someone else,that there will be some sanding and re-sealing involved.
Now,because this method was shared with me before the Eraser came along,by a cuemaker that was a member here but is no longer with us,I will share it because it works,and hopefully helps you.
There is a very common yellow kitchen sponge that comes with a green Scotch Brite pad on one side available anyplace that sells dishwashing products. Ignore the green side,you'll only be using the yellow.
Take one of those and wet it,then wring it out as best you can.
Microwave it until it's steaming,then go to work.
I never tried it by hand,but using it on a lathe works best if you take about a minute to go from one end of the shaft to wherever the clearcoat ends if there is one,then turn the lathe off,rotate the sponge to a clean spot,then go end to end WITH the grain,and wipe dry then inspect.
Only a truly noticable dent should be left. Apply whatever sanding and sealing is needed. Tommy D.
