By now, compareGroups package does not provide any specific function to know whether a chisq.test or exact fisher.test has been performed. Alternatively, you can write an instruction to answer to your question. For example, for the regicor data available in the package:
>cg<-compareGroups(year~.-id,regicor)
>any(chisq.test(cg[['Sex']]$prop)$expected<5)
Hope this helps.
Isaac.