With a little creativity

You can make a left click on the launcher icon behave as a right click.

You mention you have five applications availbale on right-click. I assume you created or edited the quicklists (desktop files) yourself, and added the application shortcuts. I also assume you do not need or use its native command anymore, since it would result in a strange behaviour of the launcher; launching one application and at the same time offering a menu to choose another...

The solution I suggest uses the desktop file's native command to simulate right clicking on the icon.

The solution

The solution is icon specific, so not the general behaviour of the Unity Launcher will change, only of the icon of which you prepared the .desktop file.

Install xdotool , (it is in the repositories) with which you can simulate keyboard and mouse actions. Then replace the desktop file's (first) Exec= line by: Exec=xdotool click 3

Then, when you click on the icon in the launcher with left click, it behaves as right-click and shows you the quicklist items you created in the .desktop file.

Cosmetic difference with a "real" right-click

Usually, when you start an application by (left) clicking on its launcher, it is blinking for a few seconds, as a "time out" for the icon before you can reuse it. Although you can use items from the menu immediately, before you can reuse it, the icon has a time out of a few seconds just like it would have when starting an application from its native command. That would be a bit annoying if you would like to quickly start more than one application from the icon's shortcuts.