I did a closer, although quick, look in my pile of schematics for the ST(E).
If it is good or bad news I don't know, but I know that I'll grab a few MC1377 if I see them as a spare part.
I seems that the STE is indeed equipped with a MC1377 from factory, which means that the composite you are looking for is likely to be present on an STE. A quick look in my own STE showed that it is present inside.
The ST is using something else, and thus seems to be handicapped in this area.
So, if you find a cheap STE, it might be what you need.