Hi,
I tend to agree with the others who feel this is a bad idea.
Is it technically possible, maybe. Though the fact that you have expressed immigrant intent, but are asking to enter as a tourist IMO could raise questions, concerns
or complicate things. Immigration officials are never the nice guys, full of understanding and tolerance from my experience.
I really don't see the problem with simply activating you GC on the way in; though it could inconvenience other people if you are travelling in a group.
Once you have entered and activated your green card, they will send it to the US address you provided for this purpose. They will also stamp you passport with a
temporary GC. In this regard, you can leave the US as you like.
It's your choice whether a couple of bucks, or short delay at the POE is worth making sure this process goes right or not.
Like i said, this may technically possible, but it certainly isn't the kind of risk i would ever be taking, no matter how compelling the advice given on a forum.
AJK