Usual non-constructive mathematics leads to some paradoxes (e.g. the Banach-Tarski paradox), which are directly related to the axiom of choice. In non-constructive mathematics, the axiom of choice (as well as proofs by contradiction) are not accepted.
For statistics, the axiom of choice implies that for any $c$ there must be a model such that the BIC (or also the AIC) is below $c$ (see this answer, we can arbitrarily increase the likelihood without increasing $N$, the number of parameters), which kind of defeats the purpose of the BIC as a model selection criterion. This method would not work in constructive math, because the space-filling curves or bijections don't exists in constructive mathematics.
Is there a general treatment of statistics in constructive mathematics? AFAIK a lot of parts of statistics are non-constructive (e.g. Gaussian distribution, Beta distribution, etc). It would be interesting to see which parts of statistics are different when the constructive approach is taken.