I'm trying to make sense of how a foundationalist can reject coherentism (or 'wide reflective equilibrium'). I interpret those latter terms fairly broadly, and perhaps that's the problem. But here's the key thought: surely if you have a maximally coherent belief set then you are not open to any rational criticism? Or again: if you are open to rational criticism, it must be that you can improve the internal coherence of your beliefs. Rational persuasion just is coherent persuasion -- after all, any rationally compelling argument must start from premises that one accepts (even if they go on to radically undermine one's other beliefs -- that just shows there was some initial incoherence there to exploit).
Can the foundationalist deny this? Do they even try? Perhaps they mean something different: their concern is not rational justification, but something more 'objective' -- call it warrant. Maybe there are certain foundational beliefs or priors that are objectively warranted, in the sense that they're what we really ought to believe, but one may fail to believe appropriately without thereby making any rational error (or being susceptible to any rational improvement).
A further application of this idea would be to distinguish proper proof from other deductive arguments. A proper proof will establish the conclusion from warranted premises. Otherwise, it is a mere 'rational argument' that may persuade (ad hominem) those who accept the premises, but it doesn't establish that they really should believe the conclusion, because they shouldn't have accepted the premises either.
[Two worries: (1) Is there really a theoretical role for 'warrant' and 'proper proof' to play here, distinct from simple 'truth' and 'soundness'? I'm not sure about this. I guess truth doesn't come in degrees, whereas warranted credence presumably would. (2) I'm not sure whether this relates to the distinction between ideal vs. non-ideal rationality. It appears independent, I guess, since even 'perfect internal coherence' is an unattainable ideal; but I feel like there's some important connection here nonetheless.]
P.S. Of course, this isn't how foundationalists have traditionally portrayed their view. They talk of rational "self-evidence", etc., but if they just mean that they assign near-certain credence to a claim (such that it will readily swamp almost any contrary beliefs) then (again) the coherentist can accommodate this. In fairness, I guess a 'self-evident' claim is meant to be one such that anyone who understands it can thereby see that it must be warranted/true. But how could this be, if not that its negation is incoherent?