Re:
It doesn't necessarily follow that the play on the inside of the cup means the outside of the cup (i.e. where the thread is) is bigger by the same amount - it could be that the cup is just 'thinner' - if you see what I mean.
Can you just use the alloy cup from the old BB? If you can it's probably the way to go - the splines on the plastic ones can break a bit too easily when you try to remove them which can be a pain in the ar*e.
If not just grease both the threads on the BB cup and the frame (clean the latter with a rag / toothbrush first to check there's no crap in the thread) and gently give it a go and see if it goes in cleanly. Don't force it...