Recursively presenting or even enumerating all P-hard languages
After some further thought, partly triggered by writing a long question, it seems clear that the set of P-hard languages is not countable, and therefore not recursively presentable either.

To see why, consider languages $L$ that satisfy $w\in 2SAT$ iff $ww\in L$. 2SAT can be log-space reduced to any such language by definition. However, there are countably many words that do not have the form $ww$, and the languages might differ on their membership status. Hence we obtain uncountably many P-hard languages of this form.

Moreover, it does not seem possible to recursively present all decidable P-hard languages either, since there is no effective enumeration of all Turing machines that are guaranteed to halt (this seems to be the minimal requirement for applying Schöning's construction).