This seems more like a Photoshop project.
I’ve never considered this challenge before, but it seems that there are a few caveats..
Initially, it seems that you would just need to shrink the image, then when it is stretched it goes to the right size.
How do you get the right scaling?
It seems that you would have to do an actual production test, with the printed stretch material and the ring.
But this idea seems to leverage the expectation that this device will evenly tension the material, which it might not.
Are you seeing an even stretch or is it distorted, such as the center not stretching much but the edges stretch greatly?
If that’s the case, in PS you could try Filter > Distort > Spherize. You could save this as an action, then batch process your files.
Otherwise, if you need more granular control, in AI you could try Object > Envelope Distort > Make with Mesh. You’d have to test the points setting until you find a setting that works for you, but that’s an option. I’m not sure how you might automate this approach, however.
Just a few thoughts.
Vince
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