I would recommend trying to find ANY other instances in your school (or any other schools/buildings in your district) where private school students ARE allowed to participate in a school sponsored activity, whether it be a club, community event, or even a prom (many schools I know of allow outside students to come to such events, after clearing a small bit of red tape).
Finding an existing example can always help with a “if this other group can, why can’t we?” argument.
The other thing I might bring up in addition to the “we all pay taxes” argument mentioned in an earlier post here, would be to ask: If homeschool students are allowed, how is allowing private school students any different?
The school our team is based out of is a “Career Center” where 11th and 12th grade students can attend for a half-day session from a variety of different area schools (in-district, out-of-district, homeschooled, and private school). Due to this reason, we are in turn allowed to draw members from virtually anywhere, regardless of the school they attend (in fact they don’t even have to attend the career center either, and can be at any high school grade).
Now, in order to attend any events with the team, each student must get a permission slip filled out which also has a section for their home school to give authorization to attend (so if you have bad grades you won’t be attending regardless of what school you come from) which has to be signed by a principal or counselor.
One of the advantages to this system is by allowing outsiders to be part of our team, see our facilities, and interact with the teachers, many of them will, in turn, choose to enroll in one of the programs at our school as well. In that way, our team can act as a feeder for the school itself, which the administration loves, so we get their support.
Perhaps a system like this could be proposed to your school board (or whoever), where you can allow outside students but it requires some additional permissions from the parents, home school, etc.
Anyways that’s my thoughts on the issue, hope it helps, and hope you guys can get this resolved!