The task calls a script programmer, but basic level Photoshop users can record an action which removes a single color from a single layer at a time. The action should contain the following operations:
- Select > Color Range
- Go to the next layer
The action should be defined to start with a function key. Press that key as many times as you have layers and colors to be removed.
Before removing one color pick that color to the foreground color and select in the layers panel the top layer.
Hopefully you live in a place where the keyboard contains the brackets [ and ] as single keypresses. Command "go to the next layer" needs it and holding the Alt key at the same time. If getting a bracket needs alone the Alt key you cannot use the command. A workaround is to redefine the computer keyboard localization to US or UK. The shortcut for "Go to the next layer" is uneditable.
If your frames happen to have some antialiasing or the borders between the colors are fuzzy for some other reason or there's some noise you may need to insert some fuzziness to the color selection and expand the selection 1...2 pixels by inserting to the action Select > Modify > Expand > 1 or 2 pixels.
As said, a programmer could write a script for the job. Your problem is so common that there exist web services which change and remove colors of GIF animations. Their processing fees vary.
To avoid the problem in the future make your animations in a multilayer frame capable program. You have some already if you are a paying customer of the full Adobe CC set. Make the GIF or other distributable format file as an export.
Just in this case you probably could purchase the wanted transparent version from @ilsoestrator. I am not at all sure artists allow unauthorized modified versions of their works.