I have lots of questions about the ownership of the theater, the operational agreement between the charity and the for-profit youth program, and who actually paid for the system.  But the underlying issue implied by your question seems to be whether the charitable community theatre group may pay money to buy a lighting system that was already purchased with the proceeds of a charitable contribution.  I am assuming that you issued a gift receipt and the donor took a charitable contribution deduction for the donation.  Otherwise the donor would have given the funds directly to the for-profit.

In the absence of some prior agreement, I think you have the much better side of the argument about ownership of the system.  If you are in a state with an activist attorney general, you might get some help in pressing your case.  You might also want to consider whether taking the dispute public would put some pressure on the for-profit.   The amount is obviously too small to litigate.

If you cannot negotiate a good settlement, you can always spend some money to resolve a legitimate dispute.  Even if you paid the full $8000, you aren’t likely to lose your (c)(3) exemption.  It sounds like too great a payment to get back what appears to be rightfully yours, however.