Zahavsdad, vouchers which can be used for religious schools are constitutional under certain conditions (Zelman v. Simmons-Harris, 536 U.S. 639 (2002)):
1. The program must have a valid secular purpose,
2. Aid must go to parents and not to the schools,
3. A broad class of beneficiaries must be covered,
4 The program must be neutral with respect to religion, and
5. There must be adequate nonreligious options.
This might open the door for them to be in acccordance with Blaine amendments.
As for askanim acting on behalf or or in coalition with other religions, under the US Constituion it is all or none. Thus Rav Moshe allowed the Aguda to act with another religious group to lobby for Federal programs that would benefit yeshivot (leaving aside federalism questions).
So far as participating in advisory committees, there are some who interpret Rambam as REQUIRING us to try to convince non-Jews to follow the Sheva Mitzvot. Even according to those who do not require it, it is certainly praiseworthy. Again, this requires participation of other groups but there is no reason why the Orthodox representatives cannot disagree with them.