Low-ish Budget Wavy Gold Gender Neutral Engagement Ring - Options?

hi everyone!

I'm looking at proposing in about six-eight months to my wonderful girlfriend. We talked recently about what we want to do about rings and we want to pick out our own wedding rings together (looking at vintage only) and they're not planning on stacking their rings. They want me to use a low budget ($150-300) but still nice ring that they can use as a "travel ring" after we get married as an engagement ring, which I'm thrilled with, but they want me to pick it out!

My girlfriend has a very interesting sense of style. They only wear gold, which narrows it down, and they LOVE shapes and squiggles. They're currently on the hunt for a Seiko Half Moon Watch and they wear these Squiggle Earrings every day. I would love to be able to find them a squiggly ring that isn't really cheap but also doesn't break the bank. It doesn't really need to scream engagement ring, but I don't want something too casual. Stones optional, but if it does have them we'd want them to be small, maybe even flush set. I'll go to a local jeweler to work something out if I can't find anything I like, but I feel like this is kind of weird and I'd love to have some better reference photos when I do!

Does anyone have any ideas? Here's some links to some rings I've seen and liked but were either too under or overpriced or just have issues one way or another:

Jared Wave GLDN Flow Brilliant Earth Mobius Atolea Aphrodite Elaine B Aura

Thank you so much!!