SPSS can’t calculate Exact test for 4x7 table of my data so I did it with Monte Carlo. The output is given below. Considering the rules of Chi-square, I should use Fisher’s $p$-value for the result of my data.
The whole point of the statistics in the SPSS Exact Tests module, which includes the Exact and Monte Carlo results, is that they don't rely on asymptotic approximations, but either directly calculate (exact) or estimate (Monte Carlo) p values instead. So if the rules to which you're referring have to do with when you can use asymptotic approximations, they're irrelevant, and the answer to your question is yes.
Note also that although sometimes you can't get the exact p value calculated using the Exact method due to memory issues, it is possible to get very close to it by extending the coverage level of the confidence intervals for the Monte Carlo method and running a huge number of simulations. If you run enough simulations for many problems you can get a 99.99% interval to have the same lower and upper bounds to 2-3 decimals, which is typically plenty of precision for a p value.