Hey, you're basically me five years ago.
Legato guy, huge Satriani fan, and pretty mediocre alternate picking that I compensated for by just really leaning on my legato technique.
A couple thoughts:
1) At the end of the day, if your legato is *that* good, it doesn't really matter. By all means, practice stuff that you're weak on and try to become more well-rounded, but I've been much more comfortable with who I am as a musician from the point several years ago that I realized a player's style is as much the things they CAN'T do as the things they CAN. But, if you want to get by on legato alone, it has to be pretty damned good.
2) Definitely focusing on keeping your picking hand and arm (and, for me, I found I was often tensing up my shoulder while picking, so I'm trying to keep it relaxed consciously when I practice these days) loose. What you're seeing in Morse's playing may not be "wild" picking, so much as fluid and relaxed. I've found feeling like there's a bit of "snap" in my wrist helps a lot, especially with articulation and "energy" in a part.
3) There are two resources I'd start with, for alternate picking advice. The first is this awesome Paul Gilbert video - Gilbert is about as good as anyone has any right to be, and happens to also be an excellent teacher:
Second, Troy Grady's "Cracking the Code" is actually really, really good. Here's the first episode of Season 1 - he doesn't really start diving deep into mechanics until Season 2, but Season 1 actually lays a pretty solid groundwork, while also just being really good entertainment - I think I binge-watched the entire first season over a couple drinks one night, jamming along, and it just really took me back to the feeling of being a teenager, and hearing someone like Satriani or Vai for the first time and just going, "What the f....?!?!?" because I couldn't even comprehend what they were doing.
Also, don't feel bad if you're struggling playing Steve Morse parts. The guy's DAMNED good.