About the argument: "she's paying for it anyways, whether the bin is full or not".
If you look at it from the opposite direction: She is not saving anything by "sharing" it with him.
If he's using it only a few months a year, I would suggest contributing a third of the cost. Saves him the transport and hassle, and she benefits from sharing. I agree with previous comments that if he wanted to be nice, too, instead of just the OP "being nice", he would have at least offered a gift or offered to pay a part; so far that has not happened, and the OP says it's been years.
OP does not lose anything by not sharing, either, while he does not contribute.
The neighbour could also have offered to take her garden waste to the tip along with his own, saving OP the full cost (and probably she would give him a thank-you gift or petrol contribution). That has not been suggested by him, either.
If they don't have anything else where they trade help/favours, when it would balance in the long run, then I'd ask for a contribution, depending if you would share half of the bin or a third (seeing as he does not always use it). The 20 could go towards a supermarket shop or anything else, neighbour saves the trip t the tip, both benefit from the share.
But if you fear your bin will be full right after collection, as he has history for, then definitely change something about the situation or that he can only use it before the next collection when there is space.
From an environmental perspective, it would probably be best he has his own bin, the collection truck will go through the street anyways, but if he transports tit to the tip, it's an additional drive.
Financially, it seems the best option is to share the cost. Then both benefit.