The developer just want her to be a boss fight but not because she is equal to Dante... That boss battle cannot be take in consideration... Even jaster boss fight can put you in danger but this doesn't mean he is equal to Dante, since jester is probably the weakest demon in dmc 3... Cerberus himself can kill you easyli but we can clearly see in the after boss baltte cutscene that dante didn't even brake a sweath to defeat him... If gameplay had to be taken in consideration than dante should be able to destroy every weak enemy whit just a single shoot or he should be able to destroy every wall whit his sword... Yamato should be able to cut trought everything and kill every enemy whit just a single hit but that doesn't happen for obvious reason, gameplay it's meant to be played, some part if not most part cannot be taken in consideration while talking about story and character power...
Yeay lady shoot dante in the head but dante didn't even care about that, he simply didn't think that lady would have shoot him because if she shoot him she would have fall, thats why he didn't se the shoot coming... Dante simply took the shoot but doesn't care about it, he could have taken another thousand shoot in the head... After the shoot dante was upset because he was trying to save lady and she shoot him instead of thanking him, not because the bullet was a treath to him...
The fact that dante and lady bullet touch each other is probably because dante shoot the bullet in a manner that they cannot kill lady but only touch her oncoming bullet instead...
In this cutscene is clear as day that dante is far superior to lady and only joke whit her... Lady fight to kill dante while dante is simply playing game whit her...
Also in this same cutscene you can clearly see that lady shoot first and than dante shoot at the bullet she is firing...
Also gameplay doesn't count because as I was saying before even a simple enemy can kill you in 3 hit at the start of the game, but we clearly see that dante can take many attack from enemy whitout even bothering it in the first game cutscene...
In the anime lady and trish combined power were not even able to scretch Abigail, they need dante help to kill him, so Its pretty ridiculous to say that lady power are equal to Dante or that she can put him in danger, since there is no single moment in cutscene in Wich lady put up a treat to Dante...
Also in the anime Trish and Lady fight but you can clearly see that Trish is playing game whit her and doesn't want to kill her while lady put all she have to fight Trish... So lady is not even comparable to Trish, let alone dante...
dante face off against stronger opponent in dmc 3 while lady only fight simply demon and always need dante help to defeat more powerful demon like in the anime...
We are comparing a simple human beings whit the legendary son of sparda dante, a being who was able to defeat opponent like mundus the demon king, Abigail and argosax, this last one whitout even breaking a sweat... For how strong lady can be for a human, there Is no way that he can be put in dante league, or that she can even prove a treat to him, the only enemy that were a treat to Dante in the entire series were mundus, vergil, and abigail, the last one only because dante get careless against him...
Lady is just zero treat to Dante