I don't know anything about avocados specifically but I will tell you what I know about fruit trees in general.

Prune in the winter prior to buds leafing out, at the end of the dormant period.

Fruit appears on secondary or tertiary branches, usually tertiary on most fruit trees (your trunk comes from the ground, primary or scaffold branches come from the trunk, secondary branches come from primary branches, and tertiary branches come from secondary branches).

So if you prune off all your tertiary branches you will have no fruit until the tree can regrow them.

That being said, production trees are pruned all the time to maintain a managable size. So I'm sure you can prune it, even severely. Sometimes pruning a tree as such will reduce the visual appeal of the tree, so it isn't done in the home setting often. But in an orchard setting it is done all the time. Just make sure you leave sections young enough to sprout new growth.