I created Button1 and assigned the Enter key to send Alt-Enter in its click mechanism, as you suggested in the question:
Sub Button1_onClick()
Call Application.OnKey("~", "SendAltEnter")
End Sub
Sub SendAltEnter()
Application.SendKeys ("%~")
End Sub
This does indeed re-route the Enter key, however it appears that Alt-Enter calls the procedure for the "Enter" part of "Alt-Enter" again, resulting in an infinite loop the first time you push enter after clicking the button, and you must restart your Excel application to clean it up.
I also tried it with another key near Enter, # (at least on German keyboards), which could be used instead of Alt-Enter:
Sub Button1_onClick()
Call Application.OnKey("#", "SendAltEnter")
End Sub
Sub SendAltEnter()
Application.SendKeys ("%~")
End Sub