Varnishing can often be more "art" than "science". I don't consider myself
a "pro" by any means, but have spent enough time around the pros to pick up
some of their tricks. Sounds like you're using all the main ingredients to
a successful varnish job:

* using a good brush, "China Bristle" or better, although foam brushes have
worked for me,
* never shake the can, stir -and- strain the varnish to a separate
container,
* Schooner Varnish is the preferred varnish of the pros I know,

A couple things you didn't mention though, and I'd add:

* surface prep: make sure you thoroughly clean and tack rag the surface
before each coat (contaniments can cause what appear like bubbles).
* air temp / humidity: calm, cool and dry mornings are often the best time
to work, gives the varnish more time to settle before drying.
* lots of coats: 8 coats as a minimum, most pros I know go with 12 or more.
* brushing technique: sounds like you may be moving the brush too quickly,
thus generating bubbles. Also brush in one direction, not back and forth.
Try to "flow" the varnish on rather than "brush" it on.

Of course, there are days when the varnish just doesn't want to cooperate.
The "cheat" I've seen the pro's use is using very heavy coats, and "flowing"
the varnish on, then if it still won't level properly, they'll wet sand and
polish as a final touch.

Some folks love doing brightwork, others (like myself) see it as a necessary
evil. Trust me, you aren't the first to be frustrated with a varnish job
that just doesn't want to work.

Good luck!

Paul