Oh you just gave me an idea for another hypothetical. If the House was unable to elect a Speaker during the time the debt ceiling was breached, would they even be capable of bringing any action before the Court?

One stalemate averted by another!

It is only a rule of the House that limits business that can't be done pending election of a speaker (and before the recent crisis was resolved, one of the solutions proposed was changing the House rules to allow more with an Acting Speaker), so, not in any way that the House itself couldn’t bypass, even without electing a Speaker.