I am doing something similar, so let me share my thoughts.

You might not be in the greatest position in that (a) Amex and Chase might not have the same rewards partners and (b) you're on somewhat short notice to really rack up rewards.

I got tickets to Paris for the minimum, which is 60K coach and 100K business. Actually we are flying eastbound separately, so going coach, then flying home business class together. If you could cobble together 120K or more points on Delta, and are somewhat flexible on your dates and persistent in calling Delta, you might be able to get tickets.

I also just got a Hyatt card for myself and am trying to get one for my wife. I second the idea of using the free nights at the Park Hyatt Vendome. That's what I"m trying to do. Note that all nights are not necessarily available.

I suspect you can get a better deal transferring points and then using them for flights, than using "pay with points." You will get value of 1.25 cents from Chase. It wasn't clear to me where you were leaving from, but you probably can get an award ticket at a better value than that. I would suggest, given the nature of the trip, that you try to get the extra points and fly business class, if you have the time to collect them first.

You might check out the "coupon connection" on this site. There are sometimes people offering to trade points, so you might be able to get the Chase Hyatt cards for your hotel stay, and trade your UR points to someone for Amex points deposited to your Delta account, then fly on Delta. A lot of people like to fly the Airbus A380, and Air France (Skyteam partner) flies that from IAD to CDG.

Keep in mind that if you do stay at the Park Hyatt, you will not want to use, say, 4 free nights and then pay for the 5th. The cheapest room are $700 a night.

Also consider using Priceline, rather than rewards. For example, a 4-star hotel in Paris will be about $300 a night from the hotel, but $150 a night from Priceline. That may well be a better value than using points.