Ah, I see! Well... if now your updates allow HDR in the shader and HDR in the STM inspector, the only place left I believe should be the mesh itself...? Apologies for debugging this way, I tried setting up my own scene with post-processing and wasn't able to see the same issue... bloom seemed to apply to everything in the scene? Sorry, I'm really new to using post-processing.
Inside of SuperTextMesh.cs, look for "endCol32", "startCol32" and "midCol32" and add "[ColorUsageAttribute(true, true)]" above where each of these is defined. I'm not 100% sure this'll work but these variables get assigned directly to the mesh itself so maybe that's where it's forgetting the HDR values? I'd like to support this so hopefully this is the solution.