Should I use Automator or another program for repetitious tasks?
It looks like the TexturePacker program includes the ability to process images from the command-line, which you could then batch process with a script. You'll first want to install the utility by clicking Install Command Line Tool under the TexturePacker menu in the app.

Then I'd suggest having a look at the documentation, which includes the command-line parameters for most of the options in the app. Have a look at that, and see if you can get  the actions you need on the command-line. If you need help turning those into a batch processing script, I'd suggest coming back and amending your question with the specifics.