Polly-Shades has the stain and the urethane of sorts already mixed into the product , i just used a brush made specifically for stain, yes if it dries on you while applying you will have a hard time, I think dude said it right "wet edge"
The blush is precisely because you're sanding. You're taking the nice, smooth, shiny surface and putting millions of tiny scratches into it.
You have identified your own problem: As you indicated a slower application rate allows you to eliminate the bubbles and that eliminates the need to sand, but it's drying to fast for you to work slowly. So, you need to ...
I prefer a decent foam brush, but I don't see why a pad couldn't do as well.
Since you have a large area, you will need to get the finish on and leveled out and uniform over the entire surface as quickly as possible, at least for the finish coat. To get real time practice, using a foam roller, get the finish over the surface uniformly, after that is done, ...
well i just got done with staining and polyurethane a few projects..
i was supposed to sand smooth and coat but i just hit it with 60grit and brushed it on.
there are stain/poly type products that are applied with one step..