There are a couple of solutions to meet what you're looking for
Assign Mouse Button
If you mouse has more than the standard two buttons (left and right click), you can assign one of them to Launchpad.
For example, on the Logitech MX Master series, there's a "gesture" button that you can press with your thumb. Using the included software/drivers from Logitech (Logitech Options), you can assign the buttons keystrokes like F4 to display the LaunchPad.
If your mouse doesn't have this type of software, there are a couple of 3rd party options you can use
USBOverdrive (Shareware; $20) - This will allow you to customize the buttons, wheels, switches and controls to custom program what they do - like assigning a keyboard shortcut to them. You can use it for free but with the "annoyance" reminder. However, the product is good and worth the $20 for what it does.
ControllerMate (10 building blocks Free; $30 Commercial License). This one is very powerful and you can control basically anything that has a switch - mouse, keyboard, switch box, MIDI device, etc. You can create simple workflows like sending a keyboard sequence (if this is all you do, it will be free) or super complex workflows where the press of a button will do any number of steps (I personally use this)
You can "move focus to the dock" (the technical term) with a keyboard shortcut. Settings → Keyboard → Shortcuts. The predefined shortcut is ^ ControlF3