Of course it is a complex issue. 501(c)(3) status really only pertains to an organization's tax exempt status. It is America's civil right's laws that deal with discrimination. However, the courts have been very leniant regarding "discrimination" whan done in the name of "religion." I know there exist in America, 501(c)(3) religious organizations that have prohibitions against women holding certain offices.
There is a narrow line between allowable doctrine and tax exempt status and discrimination law.
be a religious college and refuse to allow an Atheist to teach religion (the United States V Holy cross college) But you cannot
be a college accepting federal funding and do the same thing!
You can refuse to allow a gay choir director, or even a women pastor, but you cannot do so in a private school accepting federal funding
Now here's the catch. The public school system began selectively discriminating against Christian gatherings in favor of Wicca, and Muslims and many others even after a Supreme Court ruling banning selection of student instituted gatherings.
So you have very defining rules mostly centered around the acceptance of public money or activities on public grounds.
The courts will most likely remain lenient in that regard. It is almost impossible to separate doctrine from religious freedom. The laws and what constitutes "harm" will remain flexible-but I suspect only in regards to western Judeo/ Christians sensibilities. I don't think you are going to see Shia law respected and protected under 501 status here anytime soon. Law isn't perfect; it’s a guideline of accepted community standards of behavior.
So an aikido club -as a non profit- will not enjoy the same freedom-or protection under federal discimination standards as it does not have a "belief system" covered under the adjoining freedom-of-religion laws. Worse they may be treated differently under the law, if the same aikido club is meeting in a) a church or b) a Fed. funded building of some kind then an equal 501 group (like a Baptist bible study group) renting the same space.
Confusing? You betcha.
As for women in the dojo. I have no trouble with the OP. I train with women all the time in my dojo-I just make sure to shower after.
The problem is...
I can't get them to join me!