I’m pretty sure they’re both “correct.” Personally I’d tend to use “more clever” because I don’t like the mouth feel of “cleverer” but that’s a personal preference (hmm, and actually I don’t like either so I’d rephrase to use something else if at all possible) but Google Ngrams suggests that “cleverer” has always been more popular.
If something can be cleverest - which it clearly can be, “this is the cleverest post I’ve ever written”, e.g. - ipso facto it can be cleverer, too. It’s just that, as Josh says, when you try to say the word aloud the second half can dissolve into Rural-Juror style mush, so preferring “more clever” seems perfectly reasonable too.
Obviously using the French syntax with an extra word shows that you’re more clever than the peasants who use the native English syntax with a single word. That’s the whole point of using French syntax!
Historically, this use of “more” arose in imitation of French plus, and there was a tendency to use “more” with French-derived words and “-er” with English-derived words. That suggests it should be “cleverer”, since “clever” is an inherited English word. But this tendency has been well and truly confused over time, and now it’s mostly arbitrary which adjectives accept “-er” and which don’t.
(Latin once had a cognate to English “-er”: brevis “short” > brevior “shorter”. But in French this got eroded away and mostly disappeared.)