Application customizers currently provide 2 places to inject your display elements, one at the top of the page, which is right under the Suite Bar in your image, and one at the bottom of the page. There is no option to add/change anything in the Suite Bar as part of an SPFx solution.
It is possible to do DOM manipulation if you really need it in the Suite Bar, but that is not a recommended approach since Microsoft can (and does) change the Identifiers and Classes for Office 365. If they change what you are using in your selector, your solution WILL break.
Check out the Extension Samples at https://aka.ms/m365pnp to get ideas on alternative ways to accomplish your result.
Microsoft MVP, Office Development
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