The curves appear to be almost exclusively decorative and providing minimal support to the structure. Having said that, I would edge join enough boards to create a pannel with a width slightly larger than the curve and cut it out on the bandsaw. Ti get four identical pieces, you have a couple of options. 1. Stack your four panels with double sided tape and cut all four of them at once on the band saw. Or option 2. Cut a pattern from hardboard or similar material. Stick it to a panel with double sides tape, cut it out on the bandsaw, then use your router table with a pattern bit to trim the piece to the pattern exactly. I would prefer option 2 because I am comfortable witht this method. The trick is to take the time to make a good smooth pattern. Its worth it since you will likely have it forever.
1) Steam bend;
2) Laminate for the bend;
3) Glue to the top and bottom of a 2x, then cut on the band saw or with a jig saw, which would be the easiest of the three methods. I just chopped a 4x in half, then chopped pieces off the end of each and added them back to grow it from a 4×6 to a 4×10-1/2, to make legs for my lathe stand. When done, they were, somewhat, like heavy Cabriole legs.
I believe this does add to the strength of the end product. The two supports work with each other. While one is fighting lifting, the other is fighting the table top from going down.
If you are just painting then I agree with Lanwater just glue a couple sheets of 3/4 plywood together. If you space them right I bet you could get away with a 4′×4′ square of plywood. If you want to stick to dimensional lumber just laminate 2×6 or 2×8′s and then cut the curves.