The Constitution says nothing about a budgetting process - the dent ceiling included, so can always be modified, and is from time to time. Mitch McConnell, the Senate Minority leader, has proposed a change in order to avoid the looming default.
The story is detailed in the Washington Post and Politico, and it is a but complex, but it allows the president to submit to Congress a request to increase the debt ceiling. Congress can pass a "denial of request" which coudl then be vetoed by the president. Since it takes a 2/3rds vote to overide the veto. the president's own party can ensure it will not be overriden. This means the debt ceiling is raised, and there is a voting record showign who voted for it and who voted against it.
Obviously this fits within our 2302 discussion of the internal rules of Congress and how they change over time. Since this is driven be expedience (and this asumes the proposal passes), its worth wondering how many rules in Congress were the result of similar circumstances.
I like to make the point in our discussion of the executive that its increase in size occurred incrementally over time and generally in response to specific historical events and problems. Most recently, the Department of Homeland Security was created due to 9/11. Without that event, the department would not exist.
Are congressional procedures the accumulation of various decisions made to deal with ceratin political impasses? Might be a good study.