We know a builder would have to do that but I was hoping someone might come up with a tricky sort of clever idea that we haven't thought of.
Always good to ask and to be informed of what can or can't be done, even if it breaks the current idea of a budget. Better to know what the "right way" would be rather than only have a contractor set something up the "wrong way" because of a limited budget.
As in, they'd only sell you on an idea they know you can afford, even though it'd be a disaster over time.
Then there's the case of not over-spending the resale value for the location. While a 'done right' addition might be terrific, it'd be a shame to sink a lot of money into it if there's no chance of recouping the expense on resale. I don't know if that'd be an issue for your location, but it's definitely something to keep in mind for any kind of major improvement.