One point that should be acknowledged is that on a percentage of use versus percentage of payments basis, the air carriers do pay more than light GA. What the air carriers don't tell people is that the air traffic system is sized and scoped based on the needs of the air carriers, and the size and complexity (and therefore total cost) of the system is driven by their demands. If light GA went away, it would hardly make a dent in the cost of the ATC system (even the outlying airport VFR/contract towers are generally there because of a half a dozen regional carrier flights a day), and the amount the air carriers would have to pay would be higher, not lower, since the total cost would change little but GA's contribution would disappear. Would it cost any less to operate the ARTCC's and the big, expensive TRACON's like NY, Potomac, Charlotte, etc., if GA traffic disappeared? Not significantly.
OTOH, if the air carriers went away, the cost of the ATC system would plummet dramatically. For example, all those Class B airspaces would disappear, and the traffic burden placed on the TRACON's in those areas would drop to near zero unless the weather is sub-VFR, and even then, the VFR flyers would stay home. We could cut TRACON's like NY, Potomac, Chicago, etc., to a tiny fraction of their current size, and with it, the cost of the system.
Since the system is as expensive as it is because it's the way the air carriers want it, not the way GA needs it, the air carriers should be paying a disproportionate share of the cost. GA should only be paying the marginal additional cost of what little additional system capacity is driven by our use. The current fuel tax structure provides a good matching of the cost of the system required to meet the users' demand for system capability with the demand those users place on the system -- bigger planes burning more fuel require a disproportionately larger and more complex ATC system. Since that's a good match of user demand to system cost, there is no need to alter it.
Y'all can take that and play with the wording to suit your style when you write your Representatives and Senators.