Yes, bartacking, as previously suggested, is the way to sew webbing.
Whether the Pfaff or the Rex will sew it will depend on the combined thickness of webbing as will the size of needle and thread. However, you do not want to use a leather needle to sew webbing as it cuts the fibers and weakens it at the connections, rather than strengthening it. For big dog gear (tracking and pulling harnesses) I’ve used a #19 or 20 regular round needle and bonded T90 thread on .075 flat nylon, but what you will need will depend on the thickness of your webbing (anywhere from .03 to .1 of an inch) and then quadrupled I believe you said. I’ve heard people say that those machines will sew anything that will fit btwn the needle and the dogs, but that is all contingent on what you are sewing, what you are sewing it with and how much you are willing to risk burning up the motor. It’s not the belt or the gearing that will fail - it’s the motor. You can advance those machines by hand, but going backward by hand more than a stitch or two usually results in a rat’s nest of bobbin thread.
As @HankCowdog said, the Speedystitcher is a possibility as well and what www.strapworks.com used to suggest to folks. We have, I believe, 3 of them in the leatherworking stuff…top shelf in the hand sewing box, but I hate those things for leatherwork and don’t know if there are any awl/needles in the box as well or their condition if there are any. That being said, saddle stitching is MUCH stronger overall than the lock stitch you get from a sewing machine.
I don’t know what the specs are for your project, but instead of a net of webbing to hold the object being thrown, what about a leather, fabric or woven paracord cradle/pouch as that would save having to cut the nylon to get it to curve around the ball or whatever you’re going to launch? Say cows, please say cows!!