I'm pretty certain you just need to include the .EXE in your .VSIX. A .VSIX is just a .zip file, and you can include additional files into it by simply adding them to your project and setting the Include in VSIX property to true.
To confirm, try downloading something like the Azure Data Lake and Stream Analytics Tools VSIX to disk. (I mention this one because I noticed it installed a number of .EXE's under my C:\Program Files (x86)\Microsoft Visual Studio\2019\Enterprise\Common7\IDE\Extensions directory).
Once you have the .VSIX on disk, rename it as a .zip file and unzip, or view its contents with the windows explorer. Note that the extensions.vsixmanifest file contained in that .zip has no references or entries for any of the .exe files installed by that .VSIX. Which means, all you really need to do is include it in the .zip file.
Add a subfolder to your .VSIX project, copy the .exe there, add it to your project, set the Include In VISX property to true, and I suspect you'll be all set.
Sincerely,