[mch62 (Mark)]

You could either use a sponge or a paint brush and use plenty of the solution.
After an hour or so wash it down with a hose and use plenty of water to flush all the chlorine out.
Just monitor the timber to make sure that it doesn't deform from the solution .
if in doubt use a week solution first and see how that goes.

Lucky it's summer , so just let it dry in the sun for a few days and then seal it before a new coat of flat black.

It got moldy before , most probably because flat black by nature is porous and would not have done much to seal the timber.

When I made the middle section and the rings for my Newtonian (see avatar) I actually painted all unfiberglassed timber surfaces with fiber glass epoxy resin to seal the timber.

Hope this helps as it looks like you spent some time and effort to make that tube and it would be a shame to see it thrown out.

It looks great and you should be proud of it.

Mark