Yes, I understand. Point is you cannot use the white png straight as a bump texture and a ground from a height map doesn’t really work as a solution for this use case for me.
We should put things in order first. Starting with some context. I assume you are not doing just a simple mug, are you? Am I correct when assuming that you creating an app that let’s a user customize a number of merch items to order the real thing? Or is it something different?
If my assumption is correct, I would assume you are creating the base for each merch item, based on the rules and limitations from the company/client who is producing the final/real thing, is it?
If that’s what it is (or for whatever else it is), the first thing (in my opinion) would be to choose from a method/workflow. At this point, we can imagine something around decals, a dynamic texture and/or a node material. I would (personally) leave the displacing of meshes vertices as it seems a bit too hard to master (read too fancy), but again, personal opinion.
As said by @labris I also believe the fx (using material) would be better if applied to the mesh directly. Else, the underlaying mesh of mug behind the ‘customizable area’ will eventually interfere. But then again, this is just a thought at first, not knowing about the entire scope and context… which you might want to share with us now for better input? 
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EDIT: If (and only if) you are creating a customizable merch items app, this thread might be of interest to you. Although it does not include the ‘engraving’ part. But then, the ‘engraving’ would be just something added. The rest of the process, thinking, workflow and chosen method still applies.
And then the blog created around this subject:
And a direct link to the playground (worth looking at):