Is the replacement part expensive? (I did a search of the internetwork and it looks like they’re ~$10.) You can probably spend a bunch of time to DIY one, but it might be more effort than it’s worth. Even if you find a rubber material that can be effectively laser-cut, your next question will be “is it suitable for use as a spider?”. Next you’ll have to ask “how much material do I buy?” - by that I mean, are you going to buy enough to make one spider and just hope the first attempt at lasing/machining it is successful, or do you buy extra so that you can make multiple attempts?
Better than lasing/machining might be to try to cast a new one. Cut a mold out of acrylic, weld it onto another piece of acrylic (so that it can hold the liquid rubber) and find some kind of castable rubber, mix it up, and pour it in. Of course, this also warrants weighing the costs and benefits and begs the question about how much castable rubber you want to buy.
If you want to machine it, you could try freezing the rubber beforehand. Here are a couple videos to use for inspiration… (well, possibly inspiration, possibly discouragement)