"P isn't provable in N steps" is in Co-NP. Since in general it's an open question whether P=Co-NP, there might be a polynomial time algorithm to solve it, or their might not, respectively.

On the other hand, if you 1) Showed some large class of particular problems (ones that can reduced to from [no typo there] nonequivalent SAT instances, which I suspect is most of them) weren't provable in N=cP steps IF it's unprovable, then 2) You would have shown a sufficiently large class of SAT problems weren't easily solvable in N=kP steps, 3) which would imply P!=NP (and also P!=Co-NP).

If on the other hand your whole class of problems is unprovable, then you can prove P isn't provable by your theorem, which is presumably smaller than N in this context.

So, in summary my guess interesting classes of P, your question is equivalent to "Does P=NP?", which we know is a tough nut to crack.