How something is presented is tangential, and often outright irrelevant, to its Constitutional status. You statement that the Court struck down the mandate but offered a tax as an option is simply, directly, factually wrong. They upheld it, which is exactly the opposite of striking it down.
> but both taxes are indeed at the discretion of the House, or we wouldn't have the impasse that we currently do.
No, if anything relevant to the current situation (in which the critical thing which allows it to produce a shutdown is spending, not taxes) was at the discretion of the House, we would not have an impasse. The House would dictate their will, and it would be done -- no impasse.
We have an impasse because, like any law, appropriations (whether in the form of the budget or more limited appropriations bills) must be approved by both houses and the President (or by both houses with sufficient support to override a veto), and there is a lack of consensus between the three (two colletive and one individual) actors involved, not because the matters involved are at the discretion of any one of those actors.