Can ball pythons safely climb branches thinner then them?

[Bogertophis]

I mean people use branches all the time. Especially with naturalistic and bioactive enclosures and they don't really clean them.
And securing it by bolting it down.
I am not so worried he'd hurt himself by falling down, there will be plenty of bedding and I don't put it high up.
I just wanted to know if he can fit on the branch comfortably when he's older.
And I feel like providing climbing branches is a better solution for him than not, since he will climb anyway and I guess it's better than climbing the walls were he regularly looses his grip.

BPs are not "designed" for being good climbers. That doesn't mean they don't try anyway- some do, & some are better than others. Some opportunity to climb on things offers exercise & "enrichment", but for a BP, I wouldn't situate branches so your BP gets very high up, as repeated falls can potentially do some injury, sooner or later. I have an older rosy boa that climbs a little too- so she has a couple branches. Their bodies are round, so they're not great climbers either, lol, but she manages, & better than expected.

Also, when any snake is repeatedly trying to climb out of their secure enclosure (& it's not about breeding a nearby mate) I'd be asking myself why? It may mean that the husbandry is off (temperatures or hides aren't suitable?) so time to double check before you encourage them.

Back to branches- a wide-bodied BP cannot hang on well to a single branch the way a rat snake can- but 2 or 3 branches grouped would work much better for stability & fewer falls.

My snakes (RAT snakes) live in glass tanks, so there is no "bolting branches to the walls". What I do instead is trim branches to lengths that fit exactly at an angle (from one bottom corner, diagonally to an upper corner on the opposite side) & they cross over each other in the middle somewhere. Sometimes they lock firmly together (it's like a puzzle...) but often I use one or 2 pieces of coated wire* to quickly stabilize their position (wrap where they cross each other) so my snakes can climb & the branches stay in place, securely. (*hint: you could also use zip-ties)

Yes, we DO really clean branches. And at some point, if contaminated, we throw them away & replace them. As already noted, wood is porous- they can soak up & retain nasty germs. There is a limit to what you can wash off & sanitize like the rest of the enclosure. But natural branches offer way more traction for snakes than most other options- everything has "pros" & "cons".