I doubt it's possible to deny we can have justified true beliefs that are, at some level, non-inferential. I'm thinking of something like Michael Huemer's "self defeat argument" for phenomenal conservatism.

No, they would be justified. That just is Huemer's argument, i.e. appearances are sufficient to justify belief in the absence of defeaters. He then argues denying this is self defeating. I would recommend reading his argument, e.g. see his article Phenomenal Conservatism and Self Defeat (2011).

Suppose for the sake of argument that Huemer succeeds. All this shows is that the contrary position is incoherent, that shouldn't be a reason to accept doxastic beliefs, since the notion of "well-foundedness" of belief does seem to run into the problem of criterion and unless that is somehow cogently solved, I have no idea why I should accept them.

That still doesn't give me any reason to think that we can have doxastic beliefs without solving the deeper problem.

UGADawg wrote:

Yes, it's not justified by another proposition, but why is that a problem?

Even if it is not justified by another proposition, criteria 'x' serves to justify that thing. That experiences prima fascie justify is a doxastic belief. How do we come to know that such a criteria is valid source of justification? I'm only conceding that we have justification for "x" if we examine that there are no defeaters. However, you would necessarily need a method to arbitrate the sources of justification. If you have a method, you at least need something to be true, but the issue with knowing that truth, plausibly demands that you have some method by which we can say it is well-founded/true. This is the problem of criterion.