How do I concave an image to create a 3D file for use in a 3D printer?
What you describe, sounds like you want to create a lithophane; a pattern etched or engraved on a thin translucent base material (in your case a bowl) that can only be seen clearly when backlit with a light source behind it. Apparently you want to use it for another purpose.

Special software and or scripts that transform the image to the base material exist. An example is e.g. this sphere which becomes a globe when lit from the inside. Recommending a tool for creating such bowl is a little out of scope as these types of questions become outdated very quickly as technology changes or tools cease to exist. With the provided information you should be able to find software that is able to provide what you want to do.