I’m in two minds about this one. It’s your house and they are your bills, so I don’t think he should formally contribute to household bills if you do not want him to move in.
However, it is your house. If you don’t want him there as often, stop inviting him and then maybe he will invite you to his. Obviously it’s not as nice as yours, but you want to introduce some equality into things so you’d have to suck it up. I suspect you’d prefer him to stay at yours on your own terms though, ie he is not moving in but he can pay as though he has.
If you don’t want the fire on, turn it off and tell him to put on a jumper.
In relation to the hot tub, do you and / or your kids use it without him ? He may have bought it as a genuine gift for you or because he secretly wanted one and couldn’t keep it at home who knows ? But it’s at your house and if you don’t want it on, drain it and don’t put it on. You can’t use it and then complain. Tell him you can’t afford to run it and (assuming you are happy to keep it) if he wants it to remain he must pay for power, chemicals etc.... If not and you want it gone, you can sell it and keep / split the money.
If he earns a lot and has minimal outgoings, he’ll gave a bit of spare cash. He may not actually appreciate that you cannot afford these things. Have you actually explained that to him and shown his the running costs associated with the hot tub.
However, his level of spare cash in comparison to yours is not really your concern at this stage of a relationship. Your not really “life partners” / spouses or living together, you are just dating. You’ve said you don’t want him to move in, so address the balance in terms of who stays where. But you can’t have it both ways. I do not think you should have any claim on his finances and vice versa.
The hot tub is a bit of a special case and I’d not be expecting anything for other bills not related to this.
However, I do think he is being cheap in other ways. I’d expect a level of generosity with food shops, dates out etc.., between you to account for him staying with you so much, but that’s between you two. I’d not be expecting him to fund food or outings for your children - they are your and you ex’s responsibility. So I do think he is being cheap here and should cough up more. But I do t mean he should pay for all your food, trips out etc... I just mean volunteering to treat you every now and then.
You have a tongue in your head. Use it to explain the above issues and get your relationship on an even footing. If he’s not happy then maybe he’s not for you. But I don’t think you can have it both ways and have him paying to not officially live with you.