Dragonborn are arguably considered one of the weakest playable races, as supported by the answers to my other recent question: How would adding a darkvision racial trait to Dragonborn affect balance?

It seems to me as though part of their weakness, at least, is that their breath weapon is underpowered, so in another attempt of mine to bring them up to speed with the other playable races, I am considering how increasing the damage may help to balance them, and how much to increase it.

As before, my goal here is to make them a more attractive choice for players, but not to make them too strong as to overpower them. Ideally they should be no better than a Mountain Dwarf (one of the strongest-but-still-balanced playable races) at most.

So, with that in mind, here's their breath weapon's damage, RAW:

A creature takes 2d6 damage on a failed save, and half as much damage on a successful one. The damage increases to 3d6 at 6th level, 4d6 at 11th level, and 5d6 at 16th level.

I've considered the following:

• Increasing damage by 1d6 overall: this would make it 3d6, which goes up to 4d6, 5d6 and 6d6 as they increase in level. My concern is that 3d6 would be very powerful for a level 1 character, but 4d6 isn't very impressive for a level 9 or 10 character…
• Increasing damage by 1d6 at level 3 only: this means the damage stays at 2d6 at level 1, but it's increased to a more respectable 3d6 at level 3, and the progression after that is much like the above option, although still at higher levels, I wonder if it's still a bit meagre…
• Increasing damage by 1d6 more often: specifically, adding increases at levels 3, 8-9, and 13-14 (haven't decided exactly yet), so that a level 1 character still has 2d6, but at level 3 it's 3d6, which is a bit more respectable for a level 3 character, and at level 6, it's 5d6, as per my previous option. However, this starts to increase more rapidly so we end up at 8d6 at level 16, which is on par with a fireball or lightning bolt spell, even if it's only once per short rest, that's still perhaps too strong for a racial ability.

I'm tempted by the last option, but my concern is how rapidly this makes this feature too powerful. Even races that have innate spellcasting (i.e. tiefling, triton, etc) don't have massive offensive 3rd level spells like that, so maybe the second option is best?

Given that this is difficult to assess using the tools that were mentioned in the answers given to my last question, I'm hoping there's a better way to gauge how powerful such increases would be to make the dragonborn race more powerful than it is, but no more powerful than a Mountain Dwarf at most in terms of balance. Which of my above options would achieve the sort of balance I'm after?