You can't. The browser control doesn't provide any functionality for saving PDFs. All the methods available to the browser control are listed in the IAC documentation available with the Acrobat SDK.
The execCommand is not documented, but is actually a public accessible method on the AcroPDF.dll. Well it is easy to understand that it is not allowed to access the SaveAs from clientside scripting in a browser. But the very handy SendMail where the pdf-file is put in as an attachment in an email opened in the default email-client would be very nice to be able to access.
On the client-side scripting in a web-browser we have the mailto-link that can be setup with all field filled from the scripting except for the attachment.
First of all, there is a reason execCommand isn't documented - it isn't supported and may (read: probably will) be changed/removed in future versions of Acrobat. The only commands you should rely on for a production environment are the documented ones, unless you're OK with possibly being stuck with never being able to upgrade to the latest version/security fix without breaking your application.
I'm not entirely sure I understand your last paragraph - I believe you are confusing the Arobat OLE interfaces with the ActiveX control. The ActiveX control (what is loaded in the browser) only has a very limited set of methods. The additional Acrobat OLE interfaces have a great deal more in terms of capability but these methods cannot be accessed from a browser context (and require a full version of Acrobat on the client machine - Reader won't do).