I want to do the same: Add the Certificate tool to the right hand Tools section. The only info I have found about customizing the Tools section, is about Adobe Acrobat (Paid version). I haven't found a way of doing that in Reader (Free version), but I did find a way to hide the unusable tools like Export PDF, Edit PDF, etc.
Registry info for every tool
In the following registry key, there are sub-keys (like c0, c1, and so on) containing the descriptions of every Tool.
HKEY_CURRENT_USER\Software\Adobe\Acrobat Reader\DC\AcroApp\cRegistered
The string value in tTitle contains the title of the Tool as it appears on the right hand side.
The string value in tPath contains the corresponding .aapp file that can be found inside C:\Program Files (x86)\Adobe\Acrobat Reader DC\Reader\AcroApp\ENU
How to hide Tools
To hide the tools I didn't need, I used the above registry info to find the file names of the .aapp files corresponding to the Tools I didn't need (for example, the file name of the Export PDF Tool is EPDF_Full.aapp). Then I renamed each .aapp file I didn't need to .aapp.bak in folder C:\Program Files (x86)\Adobe\Acrobat Reader DC\Reader\AcroApp\ENU
The next time I opened Reader again, the tools I had renamed disappeared after a short while :)