Assume there's no constitutional problem to the President just going ahead and ignoring the debt ceiling. There's also a market component. Simply put, the value of US Treasuries is the 'full faith and credit of the United States'. World markets believe that promise is the closest thing in finance to certainty. And in moments of extreme uncertainty investors have even been willing to essentially pay the US government to hold on to their money for them.
But there's never been a case before -- except maybe arguably during the Civil War -- when the US government has issued bonds that are somehow in intra-governmental dispute. If you're buying US government debt, what if this particular auction is of Treasuries the sitting Congress says the Treasury Department isn't authorized to issue? There would be Court challenges filed immediately. Surely, you'd prefer to wait for the ones that everyone agrees are OK. In practice, there'd surely be some inverse premium on these Treasury notes since they wouldn't quite have the backing fo the full faith and credit of the US.
At a minimum that means the government would blow a lot of money simply because of that added uncertainty. But you'd also now have what amounted to two classes of Treasuries -- the good ones and ones with an asterisk next to them. And as soon as you have that, at least some of the fixedness and clarity of what a US Treasury obligation represents would be blurred.
I wont pretend to know all the ins and outs of the financial implications. I do know that even if the 14th Amendment issue seemed viable, this question of market uncertainty would likely be enough for the White House to rule it out.
It's not going to happen. This is not how this looming crisis is going to be resolved.