As you are well aware, cutting down a fruit tree is subject to many prohibitions. You ask about pruning down which is more easily permissible.
ShulchanAruchHarav brings details here
Some Poskim rule it is forbidden to break a branch off from a fruit
bearing tree [without one of the above-mentioned justifiable reasons].
Other Poskim however rule it is ...
OU Kosher here writes as follows:
There is a Biblical prohibition to cut down a fruit tree based on verses in Devarim (20:19-20). From the Talmud and poskim it emerges that there are several exemptions to this restriction.
The prohibition does not apply in any of the following situations:
(a) the tree is old and no longer produces a kav of fruit (volume of ...