Clearly the placement of our button conflicts with Essential Client. I'm worried this will be a never-ending story, with fixes for mod after mod. I wonder if there is a more general way to inject our button in the menu.

You could use the option to replace Realms, but this is not the best solution.
I would love if you could add a option to toggle the Button on the title screen and in-game.
This would be the best way (for me), because I just want to reach the menu from in-game and it's not necessary from the title screen. Other way would be, you move the button to the right.
But on the other hand - not many mods adding a extra button to the title screen, the pool is low.

That's an interesting idea. I've tried to stay clear of adding options to ModSettings itself, but maybe I have to let go of that, and actually add a setting to determine in which screens the button should be placed... I can see how the title screen button is less useful. (Yet another option is to drop it completely, and just have the in-game button.)