Wow - you are doing a lot of work and going to a lot of expense for something your child will not appreciate or remember. I get that you are trying to accommodate a whole bunch of relatives, including the child's father and his siblings/parents, which is very kind of you, but it's causing you headaches and stress, and now a big expense.
Is there a particular cultural reason why everyone comes across several states for every child's birthday? Is this a family tradition that you cannot go against even though you do not live with the child's father?
Do the elderly people in the family have the ability to navigate a big park? If they can drive from far away, maybe they can.
I wouldn't spend $200 on a pavilion for a party that will last a few hours (or at least that's how long your child will hold out). Are you also providing the food? If not, are people bringing their own picnic lunches? I think it's reasonable to invite people to a picnic, but I don't think you can or should pay for the parking fee for a bunch of vehicles unless they all agree to meet somewhere else and shuttle in using one or two large vehicles.
Is there any other facility available for free, or just the parking fee? That might make more sense. If you want to invest in a pop-up tent that you will use again, in your own yard or at the beach and that you can manage alone, fine. You could use that in a free park and at least you'd have use of the tent for next year and the next, as well as other occasions. I live in a great neighborhood and we borrow/share tents and folding tables and lawn chairs all the time, so if you have neighbors with whom you are friendly, I'd ask to borrow before you invest.
Otherwise, I'm a firm believer in having the party you can afford, not in trying to afford the kind of party that will make other people happy. And I agree strongly with the responder below who said you are setting a precedent for what these relatives from far away will expect every single year. Unless they are likely to spend a huge amount of money on gifts, and you need those gifts because of your financial situation, I see no reason to go broke over this. Please reset your expectations, of others and of yourself, so that you can manage this, and so that your son does not grow up expecting birthday parties with huge numbers of people and high expenditures. We had a small family party with whoever was around, and after age 4, we had parties with the number of guests based on the year (5 friends when he turned 5, 6 when he turned 6, etc.). We almost always had at-home parties in the basement or the back yard (with simple and classic games, no bouncy house or pony rides). One year, we went to a little candlepin bowling alley with 7 kids - they bowled 2 games, and the place had a party room where we could take our own cake and drinks. There was a pizza place across the street, so we brought in a couple of pizzas. It was great and the kids had a ball - and the parties lasted 2-3 hours tops, and that was plenty.
Rethink this unless you have a really firm cultural tradition that you have to uphold.