Egads, what a monster!
I would do some thinning cuts now.
I'd remove that middle trunk ASAP. And then I'd do some other thinning cuts to decongest the tree where it needs it.
I'd just do "thinning" cuts now, not "heading" cuts. That is, cut branches back to where they join the "parent" branch. This shouldn't cause such a growth spurt as "heading" cuts, which leave buds behind that branch madly.

Cutting now means you'll have lots of regrowth. In this tree, likely more than you want.

But summer pruning removes branches as well as leaves producing sugars which will be stored if the leaves remain.

So removing some wood in summer "sets the tree back" a bit in that it will have less stored sugars at the end of the season.

I suggest you do minimal pruning now. But that you mark the branches you want to remove come summer. Then complete the removal -- max 20% live wood -- sometime between mid-July and mid-August.