I don't think there's any problem here. The question you want to ask is: Given a section $u$ of $E$, whether you get a consistent answer for $\nabla u$, using either frame. In other words, does $\nabla (u^ie_i) = \nabla (\hat{u}^i\hat{e}_i)$ using your change of frame formulas? I think you'll see what's going on if you work this out.

I don't think there's any problem here. The question you want to ask is: Given a section $u$ of $E$, whether you get a consistent answer for $\nabla u$, using either frame. In other words, does $\nabla (u^ie_i) = \nabla (\hat{u}^i\hat{e}_i)$ using your change of frame formulas? I think you'll see what's going on if you work this out.

Let me elaborate a little bit. I never write change of frame formulas for the connection $1$-forms in isolation, because I am always confused by whether I should be acting on the frame by $g$ or $g^{-1}$. And whether $G$ is supposed to be acting on the right or left. But I know I will always get the right answer if I think about applying the connection to local section $u$ and expand it using two different local frames. Everything always automatically works out correctly when I do it this way.