As others said, him stonewalling you is ridiculous and a sign this is just a tip of other issues.
My answer is technically neither - we got engaged without rings, and just picked out our wedding bands together so I never had an engagement ring. We were only engaged a few months so it didn't seem to matter and it turned out metal (or possibly the weight of metal) badly irritates my skin, whether rings, bracelets or necklace or what type of metal, so it ended up for the best not to get one I'm really attached to.
I now wear silicone bands that I chose. I asked my husband's thoughts, but he had little input - he's just happy that I've found rings I like that I can wear again without pain. I never had a dream ring until I found silicone bands - so comfortable, lovely colours, and mine have this woven pattern.
Engagement rings are just a leftover sexist tradition where a man has to prove he has the monetary means to take care of his woman by purchasing a high value item
The sexist tradition is proving monetary value to the woman's father or other male guardian and the father/guardian's value to the groom. This still exists in some communities where negotiations to get married still happens. I was there when my aunt went through this, it was really fucking weird, and it's really not the same as engagement rings where only the person proposed to gets a say, particularly with how it's typical now to give a cheap ring that's then possibly exchanged.
I can see arguing that the tradition of engagement rings that once only the very wealthy used to do as part of a status symbol is just businesses wanting to make more money and not that romantic or needed, but calling it sexist while ignoring the actual sexist traditions feels weird to me. I obviously don't think you need a ring to get engaged, but if a ring is involved, a woman deciding what she's willing to wear as an engagement ring isn't sexist and I'm struggling to see it as shallow unless she's unwilling to consider budget.