Get an estimate from your HVAC contractor to fix your HVAC and have him list exactly what he found as the cause. Contact your adjuster and let them know your HVAC contractor says the system was damaged by lightening and that you will be including the estimate to repair the HVAC along with the other estimates from your damaged property.
If the adjuster continues to claim the HVAC was not damaged by the lightening, I would let them know that if they continue to deny the HVAC after your contractor professional says it was damaged by lightening, you will have no choice but to consult with your attorney to review your entire policy along with any additional available coverage's you may be entitled to.
Once you give them a copy of what your contractor found, you should be fine. I do know that when a attorney gets involved, the claims move along pretty quick for the insured. In this case though, it seems senseless to get a attorney involved on such a perceived small claim.
Good luck!
M