Currently normal damage weapons are balanced such that they do equal to or slightly less damage to all enemies as their special-damage counterparts do towards neutral enemies. On the surface this seems a sound idea, but let's look at it closer...
If a knight were to carry only one special damage weapon, then they'd do effective damage against 1/3 of monsters, neutral damage against 1/3 and ineffective damage against 1/3. However, few experienced knights run with only a single special damage weapon. Running with two special damage weapons, we instead get effective damage against 2/3 of monsters, neutral damage against 1/3, and ineffective damage against none at all.
Let's compare a Levi, a reasonably popular normal damage sword, with a Combuster and Acheron: Damage numbers are first/second,third hit
Levi: 203,258
Combuster/Acheron: 292,382; 203,292; 135,185 for effective; neutral; ineffective respectively
Average damage using only one of comb/ach: 210,286
Average damage using both comb/ach: 262,352 [2/3(292,382) + 1/3(203,292)]
As we can see, the normal damage weapon is slightly outclassed in average damage by the single specialized weapon, with the tradeoff that the normal damage weapon never leaves you doing ineffective damage against any monster class. However, carrying two special damage weapons (which is generally the case for most experienced knights) leaves the normal damage weapon in the dust.
Given this, I feel that normal damage weapons should be situated somewhere between the neutral and effective damage rather than slightly below neutral damage. Since you'd still get the benefit of only using one weapon slot and not having to switch weapons in the heat of battle, I'd suggest placing it at 1/3(normal)+2/3(neutral), which would be about about 232/322 for the Levi example. For a Sudaruska/Triglav this would work out to about 265/390 based on the DA's damage.
Obviously the Levi and Suda/Trig don't have the exact same damage profile on the first(/second) vs last hit compared with the special damage counterparts, but you get the idea.
Mr. Triglav is not as strong as it seems. :( People call me the King of non mainstream. Im not happy about how normal weapons become pretty useless while carrying two special weapons.
I made a thread some time ago about monsters that are strong against every ind of damage but normal. I didnt get much support. Apparently people is happy about how the normal damage currenly works, but thats because there are weapons like Supernova and Leviathan. What im saying is that normal weapons arent supposed to be in some way cooler than the others, they like those two weapons because they have knockback and leviathan has a 360 spin attack. If there were a shadow polaris nobody would choose supernova, they would try to get both elemental and shadow polaris.
But i think thre might be a hole in your idea. Lets say DA and GF, why would you worry about getting both when you can get a Mr. Triglav and do a close-to-strong damage to every monster.