The Sons of the American Revolution was founded a year before the DAR--the DAR was founded when the SAR would not let women join.
As far as I know, the issue of women claiming to be men and wanting to join the SAR has not arisen, at least not yet.
I don’t see many clamoring to play on men’s sports teams, either. I guess women pretending to be men aren’t the activists that the ‘trans women’ are...