No. As others have said, ultimately it's insurance that you can flog it if he does runner. Which is super old fashioned but there you are! Everyone thinks it won't happen to them. But statistically it happens to some of us!
It is a gift, take it as such, not EVERY tradition needs to be binned.
I bought my DH some cufflinks a few weeks later, he still wears them. So if it's about contributing, get yo man some nice bling too!
If you want a bigger ring, have a think about your motivation, I would rather have a small ring bought with love than a massive rock I knew I'd co-financed.
But no, I would not contribute to the cost of an engagement ring, I think that would just feel a bit weird!
Wedding rings, we both bought our own (but unless you have freakishly sizes fingers, his will be bigger than your and cost more)