Gym advice

Hey guys I know this question gets asked a lot here but I’d just like some advice for my scenario.

At my local small gym I’ve seen a girl my age going regularly at the same times as me mostly and she’s honestly really beautiful. I’ve kinda got the feeling that asking people out at the gym is looked down upon but I feel if I went about the right way it would be more okay? Next time I see her there I was planning to go up to her and say,

“hey this might be really awkward and I don’t really do this but I thought you were really pretty and would love to get to know you better over dinner or a coffee, here’s my number on a piece of paper so message me what you think, no worries if you’re not interested, hope you enjoy your evening”

It’s likely we would keep seeing each other at the gym if she rejected me as our town is small and we both go when it’s quieter at night so the last thing I wanna do is make her uncomfortable and stop going.

I’m just looking for any pointers to help out or warnings against doing this at the gym as I’ve never asked someone out before soo