Sure, you can be prudent and sensible and more practical. You can discuss the ring with her and buy one that wouldn't break your budget and all. Afterall, diamond is nothing but a hyped up pressurized coal, the price of which is super inflated by artificial scarcity created by De Beers and its affiliates; did you know De Beers was banned from operating directly in US for 60 years due to antitrust suites? Why feed the trolls? It was De Beers who instituted the diamond ring tradition in US, completely made-up stuff by a commercial enterprise.
But what's fun in that?? Romance doesn't see logic and is hardly a practical matter.
This is hopefully the only time you will be doing it. So make it special. While the general guideline is that a rock should cost you your 3-6 months of gross salary, you don't necessarily have to do it. Buy anything that you are comfortable with, don't sweat too much on that front.
Now, when you go to buy it, it will help to educate yourself on what to look for. Google the 3 Cs of a diamond (cut, clarity and carat). Know that a bigger rock is worth whole lot more than sum of small rocks whose total measure same dimension/weight. Look at it in a light, do you see any imperfection inside (a black dot etc)? Does it glisten?
Where to buy will depend. Make sure you get the right certificate (google on this). Costco sells diamonds these days, I believe. It's always good to have somebody you know from diamond industry (diamond district if you are in NY) to go with you for shopping the diamond.
Now make it a surprise occasion for her. Girls have this sixth sense, they say, about when they are going to get proposed. So the bigger the surprise, the better it is. Tell her you don't believe in western tradition but plan the surprise all along. As for the size of the ring, either discreetly take one of her rings, imprint the size on a soap and take it to jewellers or just take a quick guess. The size of the whole can be slightly adjusted later; I'd rather completely surprise her than make up stories about letting her try a ring out (trust me, they know though they might act otherwise). Last resort would be to recruit one of her friends; have her burrow one of her rings.
Be original without pushing too much. Go have fun and let that "practical" side of you rest. Be crazy, be wild. Make it a story to tell to your grandsons. Congrats and I wish you all the best for your married life.