How to make shipping required sometimes but not always

Hello! I just have a "how would you do this" question.

Here is a scenario:

Let's say that most of the time, UPS shipping is offered to the average customer in the United States.

Furthermore, a subset of products in the store are too heavy for UPS shipping. In the event that even one product in the cart is too heavy for UPS shipping, the entire order is affected by Rules that alter the checkout logic so that it does not offer a shipping method. In practice, the shop owner calls the customers later on and lets them know what the exact cost of shipping will be once they figure that out. If the customer is not agreeable to the shipping estimate at that time, they can tell the shop owner they changed their mind, and the shop owner will void the transaction for the products. Not fancy, but hey.

Now, because there are scenarios where there won't be a shipping method offered at all, the store configuration is set so that Shipping is not required.
However, this creates a problem 1 out of 100 times when a customer's order/checkout queries the UPS online tool, experiences a rare timeout (which doesn't return any UPS methods), and proceeds through checkout without ever selecting a UPS shipping method (which there would have been 2 or 3 offered, had it not timed out at UPS). Again, this is rare, but it happens.

To prevent this scenario, it would be nice to make shipping required. But currently, it can't be required because of the "too heavy to ship" scenario described above, where no shipping option is offered.

So... is there a way to make shipping required sometimes (when it's a valid UPS scenario) but not always (when it's a heavy freight item scenario)? And if so, how would you do this?

I should mention, that I originally had shipping required, and then had a Rule offer a $0.00 shipping option for the freight scenario. But the client thought it felt odd to make the customer select a $0.00 shipping method in that scenario, and had us change things to make shipping not required and to offer no shipping method in the freight scenario. But now, we run into the problem scenario every so often where a customer doesn't select a valid UPS shipping method.

Any advice on the best way to tackle this would be much appreciated.

Thanks for reading!