Is there an effective material that could be used to make this head and would it degrade quickly enough to not become an obstruction itself?

Here we go with the throw-away society again. Just swish-it in the bowl once you've had a clean flush. It goes under the sink, not on the dinner table.

Sometimes biodegradable means 'not usable'. I know that I hate that trash-bags degrade now--I put aluminum cans for recycle outside and when I get them to take them in, they dump my aluminum cans all out.

I suspect your plunger would be this way too--you'd use the first head, then a year later, the other three would be junk.

BTW, I believe natural rubber is biodegradable--I don't know what material the suction cups are, but throw them away if you wish.