I know I'm gonna get a lot of angry looks from the peeps around here but...just get a laminator. Any laminator at all.
Get a Chinese laminator for a few hundred bucks. Unless it's totally screwed up it should laminate just fine and it will be a lot easier.
The big squeegee is a fantastic tool and applying vinyl with it is really fast but nothing really beats a laminator when applying laminate.
In my experience, and many will probably disagree, even a chinese cold laminator works just fine. A laminator isn't some complex of machinery, it's a few chains and a few rolls. If you can find a cheap one, under a couple hundred bucks that isn't too far away from ya, I'd just go and pick that thing up. It'll save you a lot of time and problems.