This is a layman's answer as I am not sufficiently familiar with Illustrator or related graphic editors, but this may give you a head start on the answer.
Consider to create an image of a grid of lines, or a checkerboard of squares. Apply that image to your print pad and execute a print. Obviously, the previously rectangular image will be distorted. Take a photograph of the distortions on the bowl with the camera as square as possible to the flat surface of the bowl. I believe to minimize distortion, a zoom lens used from some distance away will give the least parallax distortion.
I use Paint Shop Pro and have had to perform "undistortion" features to clean up an image.
I understand that Illustrator (and possibly Photoshop) allow for scripting. I'm aware also that Paint Shop Pro also supports scripting.
Import the photo into your desired software and manually apply the distortion necessary to "square up" the image. Note that this adjustment is the opposite of that which will be required to pre-distort your desired image.
As a test, create a new set of grid lines or checkerboards and apply your inverse adjustments, then perform a test print.
I believe that this process will result in a set of parameters that can then be transferred to a script to automate your goal.
I am envisioning a transfer press which uses a hemispherical application pad and presses in a vertical direction on the item being printed. If the process is more of a linear action, the concept is the same, but the distortions will be biased more in one direction rather than in a radial manner.