Here is my answer to David's initial question, far from being a final one
Let's suppose X is such you can "limp" (ie put in the minimum $1 required) atc.
By theory experience it's pretty obvious you should also have a raising range (you put in more than the minimum $1) and it's also obvious your raising range is going to be stronger than your "limping" one.
Now, if you can "limp" with the worst holdings in your range (may or may not be 32o, not important), it's obvious the button will never fold to a limp, because he's getting better odds than you to put in the one dollar and your range is weaker than atc, while his range was atc when you decided to put the $1 in (ie with the worst holdings he can have he's in a better situation than the one you were in when you decided to at least "limp", so he's going to do the same).
Now, since our "completing" range is pretty weak (weaker than a random hand), button is going to raise you a lot and you are going to fold with your worst holdings (unless X>=110 obviously, in which case we already know it will be a preflop allin atc).
We don't know how often the button is going to raise, but probably somewhere along (at least) 70/80% of his range, given what we know about HU play.
The remaining times we are going to see a flop and our equity is around 35%, but given the fact our hand is very weak and will mostly flop bluff catchers out of position, it's unlikely we will win our fair % of what's in the pot. I'd guess our true equity will be somewhere around 20-25%.
So, to sum it up (using 80% raise for the button and 25% equity for the cutoff postflop), if we put $1 in with our worst hand, we lose it 80% of the times to a raise and we win [(X+2)*0.25-1] the remaining 20% of the times.
Our win expectation is thus
thus to be +ev we need
(X+2)/20 >= 1
X >= 18
Using 70% raise and 25% postflop equity (my most optimistic guesses), we'd get X>=11
Using 80% raise and 20% postflop equity (most pessimistic guesses), we'd get X>=23
So I would go with something like X>17 for my best guess
X > 8 seems a little too optimistic in my opinion