: 1).. A brush leaves brush marks.

Not if the varnish is properly thinned and applied. If your varnish doesn't completely level, it's either too thick (thin it ~10%) or partially cured (throw it away).

: 2).. A roller leaves air bubbles.

Not if you tip them out with a brush. The roll and tip method is standard for applying marine paint and varnish. It's fast (important when varnishing large areas) and produces a smooth, bubble-free finish.

: I'm not talking a big, stiff rubber squeegee... more like a piece of thick
: cardboard or a piece of 0.030" thick stiff plastic .

As I said before, I can't think of a single good reason to use a squeegee of any type with varnish. If you ask people who varnish wood for a living, none of them use a squeegee. Most probably use brushes, but rolling and tipping is more appropriate for the large areas of a kayak.

If you want to try a squeegee, go for it, but I doubt you'll like it. I strongly suspect that it will be a very slow process and it will be hard to control the varnish and prevent it from running. Moving quickly is necessary to avoid obvious "seams" where your passes overlap. If a brush leaves streaks, your squeegee will, too. If you decide to go this route, I'd use a slow curing varnish to make sure you have enough working time to get the job done before it cures appreciably.