You're limited in what you can edit on the Outlook side-panel. That page, in your screenshot, is what you can add your own custom lightning components to following the documentation
However, that's where it really ends as the rest is up to you to handle in the component. Your component will do whatever it is you tell it to do but it will not tie in to any other standard/out-of-the-box functionality (like the "log email" button as you're asking).
In your use case, you'd have to create a LWC to replicate the whole process and remove the standard functionality. This entails quite a lot:
- A button called "log email"
- Present relevant fields - your custom picklist
- Present records to associate it to based on email/event contents - contacts, opps, etc.
- Creation of said Email with all its relevant info/relations
You might be better off thinking whether you could tackle this a different way - can the product information be on the parent record (versus the email) or could you just log the email against the product itself as nothing is stopping you from relating that email to a product