The basic script step you can use is Print [Restore] and you can select the printer you want when you add this step to a script you set up your button to run. Given that you may change printers in the future, you may want to set up separate print scripts for each printer:
Make a single line script for each printer you use: Print [Restore] where you specify by name the printer you want and give the scripts descriptive names like "Print to Label Printer". Then, in any scripts where you want to include a print script, use Perform Script to run the desired print script. This way, if you change a printer in the future, you just update the single line print script and all your other scripts that print don't need to be updated.
If you want a script to check the layout and use the printer appropriate for the current layout, you can write a script such as:
If [ Get ( LayoutName ) = "Labels"
Perform Script ["Print to Label Printer"]
Perform Script ["Print to LetterHead"]