UPDATE
Second idea.
To possibly satisfy any one who does not want to use the clipboard. And wants to stick to using a file (webloc)
This Applescript when saved as an Application and in the Dock.
Will when first run ask for a folder where the single webloc file is stored.
The folder should only ever have one file inside.
When run any time after it will open the file or any other file you place in the folder.
If it cannot find a file in the folder or the folder it's self when you run the app to open the url it will warn you and offer you the choice to reset the folder.
property theUrl : ""
on run
if theUrl is not "" then
try
tell application "System Events" to set chosenFile to POSIX path of (file 1 of theUrl) whose visible is true
do shell script "open " & quoted form of chosenFile & " -b org.mozilla.firefox"
on error errm
display dialog "ERROR : The URL Folder may be empty or not exist" with icon 0 buttons {"Reset Folder", "OK"} default button 1 giving up after 5
copy the result as list to {button_pressed, text_returned}
if button_pressed is "Reset Folder" then
set theUrl to (choose folder)
end if
end try
else
set theUrl to (choose folder)
end if
end run