I see that the issue here is quite similar to a previous post of yours: Why so big difference in the p values obtained from logistic regression and Fisher's exact test?
I’d suggest revisiting the answers from that post too.
@f2harrell I see that you recommended the LRT in that case. I’m curious if you think the LRT would be reliable in this case too given the size of the model compared to the amount of actual information in the data.