Hello John:
FYI - I've moved your post to another forum that might be more appropriate. :)
There are a lot of ways to potentially accomplish what you're trying to do. However, the way search engines work, they actually intend to see and consider the content on the page that all users will see. If a search engine like Google detects that the page is loading differently to the indexing bot versus what is sent to end users, the SEO of that website will be dinged.
It seems that you know that content seen on every page is deemed important by a search engine, for the very reason we might think... If the content is on every page, it's because you deemed it to be important to the user experience of your visitors. However, if this is the case, then users might be searching for it. So, you might be hoping that you can direct people to the correct page, instead of another higher-value page that might only have that content in the footer. (This is technically handled in most cases through modifying the content and metadata of the pages in question, by testing, modifying, indexing, then testing again.)
With that being said, it would be possible for you to dynamically hide a pane in your theme, based on the user-agent. You'd just add some additional server-side code to show/hide it based on the page URL.
You can use tags in the module settings for a module. When this is used, you'll again add some server-side code to the theme in this scenario. This code would traverse the module properties of the modules as they are loaded, and then modify the module's properties to show/hide it, based on the tags and the page URL in question.
If this is a chunk of HTML content that's being loaded by a structured content solution, then you can potentially do the same thing outlined above, but this time you'd do it within the structured content template, instead of the theme. In this scenario, you might even be able to handle the conditions in a more configurable way, versus a completely code-first solution.