Using a set of binary logistic regressions with multiple choice categorical response variable I have categorical survey data on people's attitudes towards a certain policy area from 13 countries. The response variable is categorical, and includes 4 distinct answers that cannot be ordered. 
I would like to build a multi-level random-intercept and random-slope multinomial model. The problem is, that the number of level-2 cases is just 13, and the model does not converge, at least not in its multinomial form.
So, as a second-best option, I am thinking about recoding the response variable into a binary form, run a series of multilevel logistic regressions, and then use predicted probabilities to show how the probability that a certain category of interest will be selected depends on my explanatory variables. This, apparently, is just a second-best option. I would like to know what the possible risks are of taking this approach, and what objections (from reviewers, supervisors etc.) should I expect.
 A: The choice between one multinomial and a series of logistic regressions is in most cases relatively artificial. Since in both approaches you select one baseline category (reference) with regard to which the odds ratios of all other categories are expressed, it usually does not matter if you have the one or the other if the reference category remains equal. The biggest disadavantage is that you cannot test simultanous parameter restrictions across the logistic models, which is rather straight forward in the multinomial case.
Nevertheless I would advise not to use random effects with 13 countries (level 2 units), see e.g. https://www.statmodel.com/download/SRM2012.pdf. 
The alternative is to use a fixed effects model, where you include one dummy per country (minus 1). The biggest disadvantage of this prcedure that testing macro-level effects is not feasible. if you don't have any hypotheses in this regard I would go for the fixed effects multinomial model.
A: I would encourage you to run this analysis in one model (in AMOS) and I do not think your data structure is problematic (see for example: Maas, CJM & Hox, JJ (2005) Sufficient sample sizes for multilevel modeling. Methodology, 1, 86-92.). 
When you run several models on the same dataset you increase the chance of making type I errors (at the minimum you will need to employ the Bonferroni Correction; which is considered to be a conservative technique).
