Right, up front here's the stats on these three weapons.
(40 damage + 1 shred) x 4 burst
20 effective range
32 ammo capacity (8 volleys)
(30 damage + 1 shred) x 6 burst
25 effective range
36 ammo capacity (6 volleys)
(30 damage + 1 shred) x 6 burst
41 effective range
60 ammo capacity (10 volleys)
All other stats are the same. At first glance, you might say "Bulldog does more damage per bullet, so it's better against armor" and this is sort of true. But this truth was never really practical and the addition of shred to ARs has made this even worse.
Suppose you are at point blank and land every shot against targets with 0, 10, 20, 30, and 40 armor. How much damage do you do per volley?
Ares/Deimos (same damage stats):
vs 0 armor -> 30 x 6 = 180
vs 10 armor -> 20+21+22+23+24+25 = 135 damage + 6 shred (this is actually just (damage-starting armor)x(burst) + (1+2+3+4+5=15))
vs 20 armor -> 10x6 + 15 = 75 damage + 6 shred
vs 30 armor -> 15 damage + 6 shred
vs 40 armor -> 0 damage + 6 shred
vs 0 armor -> 40 x 4 = 160 damage
vs 10 armor -> 30 x 4 + 6 = 166 damage + 4 shred
vs 20 armor -> 20 x 4 + 6 = 86 damage + 4 shred
vs 30 armor -> 10 x 4 + 6 = 46 damage + 4 shred
vs 40 armor -> 6 damage + 4 shred
So we've got two cases (0 armor, 10 armor) where the Ares and Deimos deal slightly more damage, two cases (20 armor, 40 armor) where the Bulldog deals slightly more damage (though for 20 armor, the advantage disappears on a second volley), and only one case (30 armor) where the Bulldog decisively comes out on top by a large margin (46 damage vs 15). However, in that singular case where the Bulldog comes out on top, it does not actually make sense to be bothering with the Bulldog. If you are shooting at something with 30 armor using any AR, you are not in a good position. Sure, I'd rather be hitting that thing with the Bulldog than the Deimos or Ares, but I'd rather not be using any of these weapons in that situation. At that point you should be using snipers or heavies, or even a shotgun would be better and the Mercy and Iconoclast have 0 shred. This is also an idealized situation where you're hitting every shot. Both the Ares and Deimos are more accurate than the Bulldog (effective range is equivalent to accuracy), and by a very large margin in the case of the Deimos.
So what should be changed? I have a few different ideas. I'm not necessarily suggesting doing ALL of these things together, just spitting out some possibilities.
1) Give the Bulldog 2 shred instead of 1. This would only work out to 6 additional damage vs armored targets on a first volley, but being able to put out up to 8 shred in one shot might give it better utility in the early game as more armored targets begin to come along.
2) Add some (5?) piercing to the Bulldog, don't remove the shred. At only 5 piercing, this is 20 extra damage against any armored target from 10 armor to 40 armor. That would make the Bulldog decisively better at close range against armored targets, though the Bulldog would still be substantially less accurate/shorter range than the Deimos
3) Put the Bulldog's effective range on par with the Ares so it can at least hit the broadside of a barn.
4) If you want something really out of the box, reduce the Bulldog to 30 (maybe even 20?) damage and remove the shred value but add a decent shock value. New Jericho strikes me as fans of shock and awe, this would probably be somewhat fitting for them. This would also make the daze immunity head mutation more useful by putting more enemies (tritons, any humans using the Bulldog) in the game that can daze you. Again, this one is a bit out of the box.
Before someone comments that the Piranha is better than all of them for x, y, z reasons - I omitted the Piranha from this analysis because the Bulldog, Ares, and Deimos are all early game weapons accessible immediately or almost immediately. It would make the early game a bit more interesting if there weren't a clear hierarchy of early game ARs (Deimos > Ares > Bulldog > Yat). The Deimos and Bulldog being different but equally valid would be a good change in my opinion.
I did not account for how abilities like Reckless, Trooper, or Bloodlust factor into this analysis. I leave that as an exercise to the reader, but if you crunch the numbers the Ares and Deimos get better relative to the Bulldog as you add more and more damage bonuses.