While I agree we’ll have to disagree, Hank gets in the bar fight while deep into PTSD due to the Tuco shooting. It wasn’t a matter of him wanting to prove he wasn’t a coward. It’s not an altogether uncommon result of PTSD. Something he tried to hide from everyone.
Hank doesn’t do that because of the Tuco shooting. He does it because he’s so badly shaken by the Cartel’s assassination of Tortuga and his fellow DEA agents that he flees back to ABQ with his tail between his legs. He has to prove something to himself, so he does. Not PTSD. Just plain old cowardice being masked by false bravado.