Lab diamonds are “fake” diamonds. Artificial and natural diamonds only differ in their level of human suffering and exploitation, with natural diamonds being higher in both.
Band metal depends on taste and costs. Platinum is the most expensive and best IMO, gold is kind of out of style(does she wear gold rings?), white gold is budget platinum, silver is cheap platinum that tarnishes.
Get a lab diamond with both your birth stones flanking it. For the band, I would go with white gold unless she wears gold rings a lot or you can afford platinum.
Also, she wants an expensive ring regardless of what she told you. She wants a giant rock on her finger when she shows it off, but should be absolutely ecstatic for anything you get. Talk to her married friends’ husbands and don’t get a bigger diamond than they got if you want to be a bro.