You can customize how your Jira looks with custom javascript or .css files, and this sounds doable in similar fashion.
(Probably by adding "visibility: hidden" to the element via .css)
A somewhat common approach to this is to have your file(s) saved on the file system and then loading them from the announcement banner.
There are a few more replies in this older thread describing the how in more detail.
However, I have not seen this particular use case so I would recommend checking that the element's id is not used in some other forms. Looks pretty unique to me though.
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