As said in the title, it isn't possible to choose the engine-casting recipe signal from the picker, and even JSON editing a combinator blueprint for force its inclusion will revert the chosen signal back to a regular [item=engine-unit] signal.

Yes, normally, supplying the [item=engine-unit] signal to a foundry will result in the engine-casting recipe, however I have a few other mods, such as those from Galore, that for whatever reason are overriding even this behaviour with their own non-fluid-ingredient engine recipes.

Is the explicit engine-casting recipe-signal picking something you can fix?

I have no idea honestly. I've never read through the mod API docs for signals, so I'm out of my element here. If you ever find out how these things work you can either post the fix here and I will update the mod, or give you permission to do that directly.