"CLS" Equivalent in BASH?

You're probably looking for the reset command.

However, the scroll-back buffer is not a feature of bash but of the terminal program. You didn't say what terminal program you were using.

Why don't you try Ctrl+l (control, lowercase "L"). This works in most shells (err terminals)...

In OSX terminal -

Command ⌘+l (command, l) leads to removing last typed command from display.

Command ⌘+k (command, k) leads to removing/clearing all display buffer.

reset (type this in terminal) leads to reset of terminal in case display becomes garbled.

not sure of equivalent in other unix flavors.

As far as I know, there isn't a way to do this any better than what clear does with bash.

I think it's a feature that could be built into the terminal you're using though. I know the Mac Terminal app has a 'Clear Scrollback' menu option (command + k) that does what you're asking for.