Suppose that someone brings up a lawsuit against your hypothetical hyper-Catholic dating site, demanding that it cater to same-sex relationships. Does the lawsuit succeed or not? And more importantly, what factor do you think will be used to determine whether it will succeed or not? If it is "these people can refuse to cater to same-sex relationships because they're actually following their Christian beliefs, unlike those people" then the Government is indeed determining what it means to be Catholic.
Let's actually go further with the thought experiment. You listed a long list of requirements for those involved in the site, including regularly attending mass and refusing to use artificial birth control. Suppose that the judge said "the Catholic Church requires that its members confess mortal sins before receiving communion, and the site does not require its members to do that, so it isn't really restricting its membership to faithful Catholics and therefore I'm requiring the site to cater to same-sex marriages." Or alternatively, suppose that the site had all the requirements that you listed, but not the requirement to attend mass regularly and thus the site lost the case for that reason. Would you consider those valid reasons to lose the case? If so, you've put the government in the role of determining who is really Catholic and who isn't. If not, why did you have such a long list of criteria? Shouldn't it be sufficient for the site to simply state that it is catering to Catholics and have that be that?