Your "stink maile" sounds a lot like our bindweed. It crawls all over plants and cuts off the light.
As for the strawberry guavas, I know they're invasive - but they're also a delicious edible. Are you not getting fruit from them? I think if I had a problem like that, I'd be happy. My advice on those is to just leave them if they're productive and cut down the ones that are really in the way. I think you're right - if you get some overstory plants going, they'll cut down the light eventually and I imagine that would lower germination and competition. Here I'm trying to grow them without too much luck yet - it's too cold! I wish they were invasive, darn it.
I tend to take a pretty lax view of invasive plants. If they're in the way or really, really bad... I knock them down. But if they have uses, I let them live. For example, we have "mimosa trees" here (Albizia Julibrissen) and we're told to KILL KILL KILL them. However, they're a good nitrogen fixer and our sand really needs that... so I grow them next to my fruit trees and plant to chop and drop them as mulch if needed. Another plant that's a no-no is the "milkweed vine." But that has edible fruit much like a chayote... so again, I keep it.
If you're really allergic to strawberry guava, I apologize.

Wish I could come over and dig up a dozen or so to plant here.