I will be using $t$-distribution to solve this problem. Specifically,the pooled variance test because both samples have size less than $30$,and both populations seem to have the same population variance judging by the sample mean which is $8$.
What confuses me is that at the $0.01$ level of significance, my degrees of freedom would be $45$. My $t$-table only runs to $30$ degrees of freedom. Am I allowed to use the $z$-table to approximate my test statistics and critical values? I do realize the $t$-table converges into a $z$-table as sample size increases.
Yes, you can go online for $t$-test $p$-value calculators as well as $z$-table calculators to verify that for around $40$ degrees of freedom or more, the two tests behave almost exactly the same. If you really want to use $t$-test though, you can use $t$-test $p$-value calculators online that can handle larger numbers of degrees of freedom. Your question says "Justify ANY procedure you use" You can say that if degrees of freedom in estimating standard deviation is 30 or 40 or more, then empirical estimation of standard deviation is stable enough to justify $z$-scores instead of $t$-test. All that $t$-test does is take into account bias in the standard deviation calculation.