Why didn't Jimmy get the permission to air his commercial?

He knew that they would turn down the commercial and turn it into something that wasn't what he wanted and would be ineffective and he would not have any control to carry out his ideas. Consider the following transcript where Jimmy watches the one commercial they had made before with his assistant Omar.

[VCR whirring] [Tone plays]
MALE ANNOUNCER: If you or a family member have been diagnosed with mesothelioma or related conditions, you may be entitled to monetary damages. Mesothelioma is a rare lung disease, usually associated with exposure to asbestos, a known human carcinogen. Long-term inhalation of asbestos fibers can cause serious or fatal illnesses, including malignant lung cancer, mesothelioma, and asbestosis.
Omar: I remember they worked real hard to get that just right.
Jimmy: To get what just right?
Omar: Oh, the... the swirl. They wanted it kind of nebulous, but not too nebulous. Then there was the issue of the speed. I remember there were a lot of meetings about that. ...victims and their families.
Jimmy: I'll bet.
call Davis & Main at 505-242-7700. That's 505-242-7700.
Jimmy: [Sighs] Cliff signed off on this?
Omar: The partners were very happy. What do you think?
Jimmy: I think... whatever happened to showmanship?

He wanted to do the commercial his way. He didn't want to get bogged down in endless meetings arguing about swirls when he had exactly what he wanted already, and he knew that they would not approve of his version so he ran it hoping to benefit off the fact that it showed results. He went with the philosophy of "ask for forgiveness, not permission".

He also explains his actions to Kim after she is punished for knowing about the commercial and not saying anything.

I honestly thought I could fix it. I thought they'd understand once they saw the numbers. I knew it would work, and it did.

Finally, unless I'm remembering incorrectly, in one of the later episodes a "fixed" version of the commercial is shown running and it has indeed been changed into a clone of the other commercial that Jimmy had criticized with the "blue swirl", confirming his fears that they would have changed his commercial into something other than what he wanted. Simply put, Jimmy is a lone wolf and doesn't work well on a team.

Although adequately answered, I wanted to highlight a key part of Jimmy's mindset:

When arriving in the office of Davis & Main, there is a switch labeled "DO NOT TURN OFF". The notice is taped in a way that you can't even flick the switch without intentionally removing the notice.

Jimmy flicks it anyway. Because that's who Jimmy is. He doesn't play by established rules. He goes with his gut and tries to do what he thinks is right.

The same happens with the commercial. There are key points where Jimmy could have talked to Cliff. At one point, it seemed obvious for him to do so. But he chose to not do it. Because he doesn't play by rules that he doesn't agree with.
And Jimmy did not agree with the rule that Cliff needed to sign off on Jimmy's hard work.

The entire Davis & Main chapter is there to showcase part of Jimmy's persona. He does his own thing not because it benefits him, but because he thinks it's right. Up until then, you could have argued that Jimmy cuts corners because it's easier or directly benefits him in any way. But he has clearly shot himself in the foot with how he behaved at Davis & Main, yet he does not regret it. Because he isn't in it for himself, he's in it for what he believes is right.
And that is a redeeming quality for Jimmy, and I think it encapsulates the essence of his character.

I also think Saul Goodman lacks this same trait of wanting to do the right thing. I expect the loss of this trait to be (part of the) key to Jimmy's transition into Saul Goodman. But this is just an estimation on my behalf, of course.