It would be great if Dynamic Trees was compatible with mods that disables getting wood with hands, requiring the player to get an axe first.

Can you give me an example of such a mod so I can see how their code handles it?

These are the ones I know, sorry I don't have links, I'm on mobile.
Better With Mods
PrimalCore
No Tree Punching

This change should be all it needs. After reading the source for the "No Tree Punching" mod I determined that it is keying off of the harvest tool for it's logic. I never had a harvest tool set for the branches because vanilla never had it for it's logs.
PrimalCore has no available source therefor it's up to An-Sar to provide this functionality if it's needed. As far as I understood PrimalCore was already supposed to be working with this ages ago.