We did. I made it a trait as I didn't want it to reduce the damage of dragons biting each other. A little awkward, but I think it works.
Yeah its a little messy. You mentioned you didn't want dragons to suffer the dragon scales against each other, but why not? That would just mean dragon fights are extra long and epic, is that really a bad thing?
Alright here's another idea, that is more applicable in regular combat.... see if the flavor works in your opinion. I'm going with two versions, one a simpler and more general concept, the second more specifically geared to thwart ranged attackers.
Dragon Scales (V1): The dragon has resistance to any attack made at disadvantage, unless the attack is a critical hit.
--Flavorwise, this represents that you have to target weak spots on the dragon or you can't do much damage. This would means that the ranged mob firing at long distance would barely scratch the dragon....though at closer range could do some real damage. It also means that the firebreath opaque effect could have some extra oomph, as you would normally be at disadvantage to attack through it.
Dragon Scales (V2): The dragon has resistance to any attack made from beyond 100 ft, unless the attack is a critical hit. (adjust range to taste)
--So this version would actually have impact on ranged spell users as well, and encourages dragon hunters to get "close and cozy" with a dragon if they want to beat them. Compared to version 1, the ranged mob couldn't get in the big pain unless they are at 100 ft, whereas in version 1 they could do it at 150 ft.
Dragon Scales (V3): The Dragon is immune from nonmagical slashing, piercing, and bludgeoning damage, unless the weapon used has been dipped in dragon blood*
*DMs are encouraged to substitute another ingredient that bests fits their campaign.
--So in this model, we are going with the notion that you need magic to even hurt a high level dragon. But for those games where magic items are low, we are offering a flavor way to bypass it. Again, this ensures the riff raff can't hurt the dragon, but allows the PCs to attack it normally.