Since I am relatively new to this, I won't pretend to have all the answers. However, I would think a license would be needed for participation in any Inex event even if it isn't a national qualifier and that needs to be verified BEFORE the race? Also, I do not place all the blame on the Fast series, as I'm sure it is very difficult to verify these things considering many licenses (or all?) haven't even been mailed out yet.
I hear your point of view on trying to grow car counts, but I just consider the purchase of a license a "must have" to race, just like a helmet or fire suit.
My "suggestion" would be as follows: If the driver doesn't have his/her license with them in hand or doesn't know the number, they hand over an application and check on the spot. If on Monday morning it is found they did apply after checking with Inex, their check and application will be shredded. If it is found they don't have one and haven't applied, the check and application will be sent to Inex and all points/payout will be forfeited for all races prior to that point.