If you plan to gift such an expensive thing on engagement than what do you plan to gift on wedding. I think that you should give her something light like may be you can give money for their (son and his fiance) dinner for two. That way such expensive plans can be arranged on wedding.
My Son Got Engaged, I'd Like To Give My Future Daughter-in-law A Gift? What Is Appropriate In Jewelry?
Perhaps a bracelet now and the matching necklace at the rehearsal dinner. Find out if she likes pearls first, if not find out what she likes and buy that. A birthstone necklace is fine too.
Hi, I think you should wait to buy an expensive gift now. Wait until the rehearsal dinner and maybe give her the pearls then. Maybe, you can take her out to a play and dinner, just the two of you. This way it will give you a chance to get to know her better. You maybe able to build a nice friendship. Best Wishes.