Im going to vote for just go outta your way to say "Hey how are you doing " thats a conversation starter. A simple " Hi" often just turns into a dead end.. When most anyone tossing me a " Hi " I typically toss back a single word , hi or hey... But ask me a question I'm much more likely to actually have a conversation.

If you toss her a " hey how are you" and she starts talking with you.. Great ! Why not ask her if she'd like to go out for a coffee after a work out...

You have nothing to lose because right now your just worried about should you say "hi" or not and all that is doing is walking yourself into a corner in a holding pattern.

The saying "Shyt or get off the pot "comes to mind or "keep cutting bait or just go fish"

Sure your going to find this maybe impossible to do , but seriously you have nothing to lose with this girl, you don't know anything about her other than she goes to your gym you have seen her here and there you say hi's off and on.

Life is short take a chance , this won't kill you I promise. Don't over complicate things by all this second guessing nonsense.

Good luck