I think I can spare 5k, but 5k can’t get you a decently sized diamond with decent cut, colour and clarity though
I guess it boils down to what you think it's decent.
Yeah what ke3bbs said. No way her girlfriends/ family will suggest lab-grown, I would put probability 0.1%. If your proposal is not going to be a surprise, then just ask your GF what she thinks.
If your GF is petite with slim fingers, 0.5 - 0.7 carat still looks perfectly great. I think if other ppl comments why the rock is so small etc. are just a**holes as everyone's financial situation is different. Unless of course, they know you are earning $10K+ a month and just being giam on a "worthless" item. If you are concerned on this part, then can follow the De Beers method 2 - 3 months salary.
If your budget is indeed 5K but want a bigger rock, I suggest you go online and check out Blue Nile or James Allen and look for ideal triple excellent. Won't get the JP light return specs, but u can find a decent 0.70 - 0.85 carat rock (make sure you research well what is a good table/depth/ etc for diamonds). G-H, VS1/VS2 grading. Lowest color to be "white"-ish is H, eye clean I think is VS2.
Too many permutations so you have to see what is your priority. Don't even need a triple excellent ideal cut, you can go even lower. See if your GF prefers a watermelon or a golden peach.
If you can go slightly higher than 5K, u can consider Brian Gavin Black series. They are similar to JP Super Ideal cuts but pay maybe 20 - 30% less. Make sure you look at scope images for light return, and look at the hearts & arrows well.
Once you get the diamond online, then go to Canary or sth to fix up the ring for $600 - 1K. At least in this way you can easily resize, and have aftercare services locally. Just be aware of upgrade policies for online diamonds (if you want to pamper your wife with a bigger rock when you earn more in the future).
That is all the research I have done the past 4 months, going down to physical stores, searching online, HWZ. I ended up like most of HWZ brothers and bought from Jann Paul as I can get the size I want for my budget and their diamonds are okay considering they are top-top tier (but some marketing la of cuz, haha). I could have saved 20% going for Brian Gavin but I get more psychological comfort buying from a well-known store in SG, get good aftercare services, and a good upgrade policy if I decide to buy a bigger one for my wife next time.
Good luck and hope this info helps.