I might suggest you post this over on the tax forum:
Taxes - Ask Me Help Desk
There are people over there whom I never see posting here - if they're here, they've been just lurking. :-) Most of the people posting here aren't even in the U.S. and wouldn't know non-profit tax laws like that. All accountants aren't tax experts. And non-profit is a specialized area. So I think you're really narrowing down your chances on this forum. (I've done work for non-profit organizations and am in the U.S. but I'm still not a tax expert, nor a non-profit expert.)
I don't think paying them rent in and of itself would hurt anything. Where the rent is coming from might, I don't know. But what you're doing could quality as non-profit as well. A lot of times people confuse what "profit" is in that context. But really... you need to go ask the tax experts. Or go straight to the source and call the IRS. They don't bite, really. ;)