As mentioned in this comment and answer How to get fitted values from clogit model, it is not clear that predicting from a conditional logistic regression model is meaningful.
It seems to me that it's even less meaningful to calculate AUC (or, worse, cross-validated AUC) from such a model. Conditional logistic regression is a relative risk model: conditional on participants being in the same stratum, this is how log-odds risk is related to the covariates. It doesn't seem to make sense to use the covariate coefficients as absolute risks, calculate expected risk, and calculate AUC. Particularly for cross-validated AUC -- since we don't split strata across folds, by definition, we're taking a model built on certain strata, and attempting to predict on other strata.
So, what tests of model performance do we have?
I'd be interested in global goodness-of-fit tests, similar to various likelihood-related tests.
But more interested in tests of model predictive ability, since the underlying research question (in my particular case) is what sorts of covariates lead to better predictive models.