She could actually round 1st, stop, then go to second. But if she did, she couldn't round 2nd and stop again unless she wanted to be called out for the look back rule.
But, as the current rule is, if the pitcher does not have control of the ball, we can go. For instance, she lays the ball down to tie her shoe.
Actually, she could stop again after rounding second base.
The Look Back Rule says that when the runner stops between bases, she must immediately either retreat to the last base or advance to the next one. Once she does advance to the next base, she has met the requirement of the rule. So, if she rounds the next base she is allowed one stop between the next two bases.
I can respect the opinion of an umpire on the ASA National Umpire Staff, but opinions are not the same as rules. In a dispute the wording of the rule is what counts.
Attempting to advance to 2nd on a walk with runner on 3rd usually only happens in younger and lower levels of play. Any good 12 and older team can defend this and will many times end up with both runners out. You will very seldom see a walked batter attempt to advance in higher levels of play.
Throwing the ball to first actually makes it far more difficult to defend because you are removing the rule from your use. With the look back rule in effect, once the batter touches first base, the runner at 3rd must immediately advance to home or return to 3rd. If you throw the ball to 1st, that runner at 3rd can get as big a lead as she dares and just stand there. If your players work the play correctly, the pitcher does nothing as the batter touches first, makes the runner at 3 commit back to the base, then make a quick throw to 2nd for the tag, and if runner at 3 attempts to advance you now have a throw to home.