A good pano stitcher will compensate for auto-exposure, but I get the client's point, as it may still be noticeable.
If you don't mind diving deeper into pano stitching, you can blend images with different exposures. Plus, the result can be better. Basically use fixed exposure, but do it 3 or more times with different settings. (Google "HDR" for more info.)
1) Find a good "medium" exposure that gets most of the pano with reasonable results, i.e., the midrange is correctly exposed.
2) Take the pano with this setting.
3) Go up 3 (or even +5) exposure steps and repeat.
4) Go down 3 (or even -5) exposure steps and repeat.
(You can either take 3 panos in series, or take 3 images in series at every angle. There are pros and cons. The former is faster and lets you use the drone pano tool. The latter gives you an additional option of blending each angle first, before attempting the pano stitching.)
The result will be over and under exposed images, but hopefully all parts of the pano will have at least 1 image that gives a good result.
Now add all the images to the pano sticher and tell it to select the image(s) with the best exposures.
The result will be a fairly "flat" image, but all parts of the image show detail. In post-production, you can tweak things to add contrast back.
Fair warning that this takes a lot of work and doesn't always work out nicely. The multiple exposures take more time, so the scene has more time to change (clouds moving, etc). And you need a windless day, as winds move the branches, so the images will not line up exactly, causing "weird ghosting artifacts".)
You'll find HDR settings in the controller menu, but imho they do not take a large enough exposure step.