I historically get confused by terms "conditional / marginal". I think it is because they are never defined in complex settings. I was under the idea, conditional would come from the model controlling for covariates and marginal would come from an empty model like yours. What are you looking for in particular?

Also, given your possible outcome distribution -- you may need to examine dispersion and see if Poisson is the best fit over say negative binomial regression or zero inflated. Also, I am imagining whichever count approach that is used, there is a "conditional" version. Meaning conditional on matching. I am not sure Mixed Modeling, which I prefer the term "Multilevel Modeling" due to confusion, is necessary beyond controlling for strata (matching) in a more basic conditional approach.

P.S., I have not done a C-C with a count outcome measure, but I believe you may need to adjust the intercept to account for the artificial balance between outcome groups, which is not likely the case in the super-population.