I think it's great when people buck the wedding industrial complex. Hope you find something really nice.
We got our nontraditional engagement ring at Mia Gemma in Penn Quarter - it was about $2k. They had some less expensive options, too. Etsy is also a good option - just stick in some price parameters and see if anything turns up that you like.
The Brooklyn store Catbird also has lovely rings in a variety of prices.
https://catbirdnyc.com/shop/home.php?cat=297&sort=title&sort_direction=0&show_all=1
And I lust for just about all of these rings - some are more than you want to spend, some less:
http://satomikawakita.com/488/rings