You could always desensitize him to the mister if you so wish.
You could do this by taking baby steps. Just introduce him to the mister and every time he doesn’t react negatively or seems comfy in the misters presence, reward with a high value treat.
Continue doing so until you could spray him with it for bath time.
But you don’t have to spray directly on him right away.
It could be a little gentle spray on the feet and then work your way up until he gets used to it.
And eventually I’m sure he will.
You could also keep him in the shower with you like u mentioned but let the stream of water flick off of you and onto him if that makes sense. This is a great way to improve the bond you already have built.
And lettuce baths are great since parrots (I’ve heard especially budgies) like to naturally lay or feel up on wet grass in the wild after a good rain shower. (This was also mentioned above)