You want to pass yourself off as an available rushee while you really aren't? You know that's lying. No,do not do that. And you would have to check the rules of the org that you want to join to see if they will take someone who initiated somewhere else. Odds are not.
You probably don't need to be in a fraternity if this is how un-gentlemanly you are.
If you don't like the chapter (please don't call it a house unless you really mean you dislike the physical structure) then drop out before I-week whether you rush somewhere else or not.
You need to drop your current fraternity before you rush another one.
I'm going to hope this is due to simple ignorance and not being a jerk.
__________________
It is all 33girl's fault. ~DrPhil