Hey there,
Could you clarify if you’re using the Jetpack Instant Search (meaning you instantly get search results in a modal window as soon as you start typing) or the legacy inline search (which loads results on a new page?).
When using the inline search you could in theory customize your results in the “search.php” template of your theme, but Jetpack Inline Search is now deprecated and we recommend using the newer Jetpack Instant Search.
With that in mind and regarding SKU, we’ve had at least one request for that before so the team might add that feature soon. We’re tracking this here if you’re interested in following progress: https://github.com/Automattic/jetpack/issues/32723
As for the shortcode, Jetpack Search indexes content from your site as it exists in the posts, meaning it captures everything as you edit it, and does not expand shortcodes. I’m going to report that to the team and get their thoughts on the possibility of getting the full content.
Finally, you could take a look at styling your search results differently if you’re interested in showcasing your products. We have included a “Product” style that is ideal for online stores: https://jetpack.com/support/search/customizing-jetpack-search/#styling-result-format
Thank you,
Filipe