I'm just answering this because the bounty is almost up and so far there are no accepted answers.
As others have mentioned above, it is possible to bake the "simulated height data" in the form of Ambient Occlusion from a Bump Map being multiplied into the BaseColor.
I made an image of a sphere as it will eventually (truly) be - smooth with no real height data.
I then took the same sphere and added "height" information provided by a Noise Texture and a Bump Node.
From here, you can "add" (multiply) that data to the BaseColor by using an Ambient Occlusion Node.
Before baking, make sure your object has unwrapped UV's As per the baking process, create a new image, un-check "use alpha" but DO check "32 bit float". Make sure the image is at least 2K (2048x2048).Load the image as a new image texture and have it selected (but not attached) in the material graph for the object.
Then, making sure you're in Cycles, under the Render Properties tab, and under "bake" Set bake type to "diffuse" (since you only need one color image), and click to de-select Direct, and Indirect influence, leaving only Color selected. I also recommend lowering the margin size. Hit the bake button when ready.
Once the bake is complete, go to the UV tab and select "Image > save" to save your image.
You can then load and connect it as a basecolor as you would normally.
The final result looks like this - note the only texture attached is the new BaseColor containing the "impression" of the previous height information: