Ignore for a moment that Hayek would have opposed the policies that created the need to borrow on a massive scale, ie: the government dissaving that drags down our national saving. Hayek's writing in Road to Serfdom and elsewhere tends to look at policy pragmatically--given the government has already enacted a sub-optimal policy, how best to deal with it, particularly without causing instability?
*this part updated for clarity*
Hayek is firm in his belief in what he terms the "Rule of Law." This is not the simple protection of property right and enforcement of law, as Hayek has already laid that out as a foundational role of government a classical liberal order. Hayek's Rule of Law is defined as (emphasis mine):
"(G)overnment in all its actions is bound by rules fixed and announced beforehand-- rules which make it possible to see with fair uncertainty how the authority will use its coercive powers in given circumstances and to plan one's individual affairs based on this knowledge. Though this idea can never be perfectly achieved, since legislators... are fallible men, the essential point, that discretion left to the executive organs wielding power should be reduced as much as possible is clear enough...rules tell people in advance what actions the government will take in certain types of situation, defined in general terms, without reference to time and place and particular people."
For Hayek, government changing a long-standing policy was a big deal because it would adversely affect people who had made individual choices based on the established rules of the game. A rule-based system is fundamental to a functioning market because it anchors expectations (emphases mine):
"(I)t does not matter whether we all drive on the left- or on the right-hand side of the road so long as we all do the same. The important thing is that the rule enables us to predict other people’s behavior correctly, and this requires that it should apply to all cases—even if in a particular instance we feel it to be unjust."
Discretionary macroeconomic policy were therefore anathema. One good example of this comes from Hayek's writings on monetary policy in the early 1930s. As Lawrence White writes,
"Hayek’s monetary policy norm in fact called for the stabilization of nominal income (MV), and thus for central bank action to prevent its contraction."
Monetary policy rules have been in the literature for a long time and Hayek has made his contribution. Given a fiat currency controlled by a central bank, the central bank should follow an announced and specific rule so that there is no deviation from expectations.
Now, the Fed pursuing such a policy in 2008 would still have led to a simple graph of M1 like this that would have bothered a lot of Rothbardian Austrians and Glenn Beck:
But as Scott Sumner and David Beckworth (and others) have spent the last few years arguing, if the Fed had an NGDP growth target, it didn't grow the money supply nearly fast enough to hit it during the financial crisis. Nor did the Fed keep inflation expectations in line with its implicit target of just under 2%-- it allowed them to fall far short. In other words, the Fed didn't live up to the markets' expectations of its behavior, and as a result we're still in a sub-optimal recovery.
So a Hayekian central bank under his Rule of Law definition would announce an explicit target of something and then consistently pursue it. (And pursuing that target in 2007-2008 would have led to policies like QE that would have made a lot of classical liberals uncomfortable.)
Hayek recognized that economic disturbances, if handled badly, caused the public to turn against free market capitalism. Hayek and Keynes were in agreement on this point. During the Depression, Hayek had argued for deflation and liquidation, but later regretted it in part because he saw the much worse political fallout.
While a staunch defender of the price mechanism, Hayek recognized that immediately removing England's wartime price controls would cause chaos and backlash against capitalism:
"(H)owever much one may wish a speedy return to a free economy, this cannot mean the removal at one stroke of most of the wartime restrictions. Nothing would discredit the system of free enterprise more than the acute, though probably short-lived, dislocation and instability such an attempt would produce." (emphasis mine).
Hayekian thought thus seems opposed to neoliberal (Jeffery Sach's) "shock therapy," eliminating any heavy wage/price controls and government interventions in the market immediately for countries that have adopted socialist practices. "Get it over all at once," as Sachs said in an interview. (One can argue that Hayek is vindicated by the observation of populist backlash against such capitalist reforms in countries like Russia after privatization was botched.)
Let's return to the debt ceiling debate. Congress has raised the debt ceiling 89 times since 1939. Judging by the yields on Treasury bonds today, the market expects the debt ceiling to be raised again and for Treasuries to keep their value. Like it or not, raising the debt ceiling is part of the Rule of Law that Hayek would expect the government to continue following. Deviating from those expectations would have consequences.
If the debt ceiling isn't raised, the President has to go through budget items line-by-line to figure out what gets funded and what doesn't. Thousands of jobs and incomes are immediately lost, interest rates rise, and the other economic consequences are quite costly. Given we already have a Federal Reserve that hasn't kept its obligation with the Rule of Law (and its Chairman indicates it's politically impossible for it now to do so), I would quote Hayek again here:
"Nothing would discredit the system of free enterprise more than the acute, though probably short-lived, dislocation and instability such an attempt would produce."
Defaulting on its debt, or meeting its debt obligations and cutting current expenditures dramatically to do so, could be as harmful to the long-run cause of free enterprise as the Great Depression was.
Default isn't being pursued or advocated by Hayekian Classical Liberals. It's being advocated by Conservatives who Hayek was very poignant about not being associated with.