How to attach a pergola to the rooftop deck

The problem isn't that there's anything wrong with PT 2x4s. The problem is that you're spanning 10 feet with, presumably, no support or bracing. A 2x6 is much more appropriate for what you're doing.

Another option would be to stick with 2x4s and run boards the other direction, maybe less often or maybe using smaller lumber, to create a grid structure. This would be stiffer and resist warpage better.

LVL lumber, like any laminated wood product, will eventually come apart if exposed to the cycles of weather. It's also not substantially stiffer than solid wood for your purposes.

(P.S. Those aren't "studs", which isn't a fancy word for all 2x4s. Studs are a vertical wall component.)

To accomplish the join I'd acquire some 1/4" by 3-1/2" by 24" steel plates and let them into opposing sides of the posts to act as gussets. I'd use countersunk flat-head sleeve bolts, piloted accurately and minimally, to sandwich the whole works.

Be sure to consider whether the existing posts are adequately anchored, as there will be substantial torque applied to them at their bases. It doesn't matter how well you attach the extensions to floppy posts.

That said, I think your plan seems flawed. A pergola creates some fairly unique structural concerns, namely the lack of diagonal bracing that a conventional roof would provide, as well as increased wind drag.

I'd be replacing your 4x4 posts with 6x6 posts that run all the way through. They'll do the job much better than spliced 4x4s, and they'll probably be more appropriate from an aesthetic standpoint, depending on your design. You don't want chicken legs under your nice new pergola.

If you elect not to do that, consider substantial diagonal bracing below the ceiling line connecting the posts to the primary horizontal members of your design in appropriate directions (where connection to a fixed structure doesn't provide support).