TL;DR: You may want to consider controlling False Discovery Rate as opposed to Family-wise Error Rate given the number of tests you are doing.
Controlling for a low family-wide error rate for so many analyses could lead to unacceptably low estimation precision for each variable. Biostatisticans deal with this all the time in gene panel studies, so they developed an alternative concept called False Discovery Rate (FDR) as an alternative way to think about the error rates.
From the link above:
The FDR is the rate that features called significant are truly null.
An FDR of 5% means that, among all features called significant, 5% of
these are truly null.
While this may sound like a p-value, note that it's a post-data assessment -- continued from link above:
Just as we set alpha as a threshold for the p-value to control the
FPR [False Positive Rate], we can also set a threshold for the q-value, which is the FDR
analog of the p-value. A p-value threshold (alpha) of 0.05 yields a
FPR of 5% among all truly null features. A q-value threshold of 0.05
yields a FDR of 5% among all features called significant. The q-value
is the expected proportion of false positives among all features as or
more extreme than the observed one.
The article I linked to also shows how FDR is connected to the Family-wide Error Rate (FWER) that you have been considering:
The FDR has some useful properties. If all null hypotheses are true (there are no truly alternative results) the FDR=FWER. When there is some number of truly alternative hypotheses, controlling for the FWER automatically also controls the FDR.
