I believe Discourse search is performed on the processed post, which contains HTML.
The search ignores html tags, and IMG tags contain no text, hence the impossibility to return what you’re looking for.
Why can’t you use the API?
You could create a local script that triggers a search query for the user’s posts containing images, iterates through the results (slowly enough to not reach rate limits, and also you can query the raw posts content if needed) and outputs the posts containing the substring you’re looking for.
Maybe there’s a simpler solution, but that’s I would go for with no other option. Fairly simple to do.