How big are the designs? It may be worth it to buy rolls of paper or extra long sheets to avoid the seams altogether. I mean, if you have the pro, seems a shame not to take advantage of the seamless cutting!
If you need the sturdiness of a thicker card stock and can’t find it in longer paper or rolls, you might want to try sandwiching several layers of the design together. I do this all the time when I need a thicker card stock I don’t have. So, cut out 3 (or however many) complete shapes, and then glue them all together. No seams and the desired thickness. For that, I really recommend good rubber cement for paper. I like the best-test brand (“white rubber paper cement”). It never causes rippling - even on very thin tissue paper - and dries clear. It’s never white at any point, don’t understand the name. Plus, you can rub off any excess dry glue and it removes completely cleanly. They even sell adhesive/glue ‘pick up’ erasers, but fingers work perfectly well. You need to rub/burnish the pieces you’re gluing together to make the glue permanent, so I’d recommend also picking up a bone folder. Except get a plastic ‘bone folder’. The bone ones create a shiny burnish that you probably don’t want all over the finished product