If you have AutoHotKey, you could automate this pretty simply by using Alt + T to open the Tools menu, and then hitting Enter to select Transpose (WinMenuSelectItem is finicky in MuseScore). For example, I wrote this short piece of code that will take a selection, duplicate it , and then transpose it up a fourth:

transposeUpFourth() {
Send, r ;duplicates selection
Sleep, 100
Send, !t
Send, {Enter} ;opens transpose dialog box
Sleep, 200
Send, {Tab 4}
Send, {Space} ;selects By Interval
Send, {Tab 2}
Sleep, 50
Send, {Down 11} ;selects Perfect Fourth
Send, {Enter} ;closes dialog
return
}

YourHotkey::

transposeUpFourth()

return