That's actually the order I do every time. I've beaten this game probably like 30+ times. You should be fine. You won't have Eagle's weakness, but he's fairly easy. He spends most of the fight just trying to knock you off. As long as you use your dash boots he's no problem.
Here's a tip. If you fire a shot in the middle of dashing, it will do double damage. This applies to charged shots as well. A fully charged shot fired while dashing will do huge damage to enemies, bosses included
Wow, didn't know that about the dash either. Just when I thought I know everything about this game. ---