I believe there actually often is a calendar association but only because the US fiscal year starts October 1st, and funding legislation needs to be enacted before then.
The idea that there's some sort of forced budgetary event on the calendar, related to the debt limit, that's then raced through to have competing solutions, then laden down with pork is Not Even Wrong, i.e. in the Pauli sense, and plainly wrong.
You raising the October date does leave open that maybe they're thinking of budgets and think the minority party in Congress has peer footing as a solution / is required to make a budget. That, at least, would explain the pork mention.