Caja does not have feature unfortunately. I had a quick look in dconf-editor too, but there's nothing for extra pane.
Seems there is already a feature request created for it:
In the meantime, you can add an extra set of commands in place so Caja will launch and press F3 for you. This creates a script in place of where Caja is normally launched, so it works anywhere (like the Places menu).
Note that this only works:
- until Caja is updated/reinstalled.
- if the Caja window opens within half a second.
First move the original aside - we will create a script in its place:
sudo mv /usr/bin/caja /usr/bin/caja2
sudo nano /usr/bin/caja
Paste this:
#!/bin/bash
/usr/bin/caja2 $*
sleep 0.5
xdotool key --clearmodifiers F3
Save and make this script executable:
sudo chmod +x /usr/bin/caja
You'll need to install xdotool:
sudo apt install xdotool
Normally a script like this should go in /usr/local/bin/ but it wasn't used when opening Caja via the MATE desktop.