UserPresetHandler.setPostCallback(function(presetFile)
  {
      // On first launch the default preset loads with no file (frontend leaves
      // currentlyLoadedFile empty), so getCurrentUserPresetName() is "". Fall back
      // to the default preset name. (Editor returns the real name, so it's unaffected.)
      var name = Engine.getCurrentUserPresetName();
      btnShowPresetBrowser.set("text", name != "" ? name : "00 INIT");
  });

I'll try it and make sure it works on first launch, and doesn't interfere with reloading DAW sessions.

	inline function updatePresetLabel(nameOnly: number)
	{
		if (!isDefined(currentPresetFile))
			return;

		local category = currentPresetFile.getParentDirectory().toString(currentPresetFile.NoExtension);
		local name = currentPresetFile.toString(currentPresetFile.NoExtension);

		if (nameOnly)
			btnPresetBrowser.set("text", name);
		else
			btnPresetBrowser.set("text", category + " | " + name);
	}

I also call this in the post save callback too.

updatePresetLabel

What does your updatePresetLabel() look like? Because I have the following already, but it's setting the text to "":

UserPresetHandler.setPostCallback(function(presetFile)
  {
      btnShowPresetBrowser.set("text", Engine.getCurrentUserPresetName());
  });

Seems like on first launch, Engine.getCurrentUserPresetName() returns "".

What I do is to force "init" in the postCallback when the presetFile is undefined, meaning it's init time...

Another way could be to save the label in preset, but this only brought me more pain down the line

Is this intended behaviour or a bug?

Depends on your script.

I use this in mine

	const uph = Engine.createUserPresetHandler();

	uph.setPostCallback(function(presetFile)
	{
		updatePresetLabel(); // Name display handled here
	});