I read in the OP "BR is obstructed on the way into 1B by F3"; followed by "as she runs though the bag "
Maybe the OP needs clarification.

If the OBS was literally at 1st, then yes, on the way to 2nd, as you and Mike said.

Re the OBS at or prior to 1B.

The umpire must make a judgement as to both where the OBS occurred and how far the runner or batter/runner will be protected.

If a fielder clearly obstructs the BR prior to 1B the the insofar as the "cannot be put out between the two bases where she was obstructed.." part of the rule is concerned she cannot be putout between HP and 1B. However the umpire may also judge that while the OBS was between HP and 1B the batter/runner would have likely reached 2B sans the OBS and in that case the protection under 8-4-3-b would still apply.

But I think it's a stretch to say that OBS which clearly happens between HP and 1B and causes the BR to round 1B wide qualifies as OBS between 1B and 2B. Ponder this: batter hits sharp grounder to F6 which skips off her glove. She chases the ball down behind and on F6 side of 2B as BR scampers to 1B inside the foul line. F3 playing a couple of feet in front of 1B and near the foul line causes BR to alter her path into foul ground. U1 signals OBS. BR touches 1B and awkwardly rounds the bag. 1B off coach sees DDB signal and tells BR (now a Runner) to try for 2B. Runner makes very futile effort and slides into waiting tag of F4. Unless judged at time of OBS that BR would have likely reached 2B I have an out here even if BR rounded wide because the OBS happened between home and 1B.

If the OBS is caused by fielder being right at 1B without the ball then I judge that to be occurring between 1B and 2B and protect accordingly.

And in that case, if the coach sends the runner into an out at 2nd, a likely award of 1st would apply.

F3 stood five or six feet up the first base line in front of the bag, in the direct path the runner took from HP to 1B. There was no way in my judgment the BR would have safely reached 2B on the hit absent any obstruction, nor was she taking a path to round the bag until after the obstruction. This is why I only protected the runner to 1B.

In the cases of obstruction while rounding as mentioned earlier, yes, I would have awarded 1B to the runner if tagged out in the same way.

It's not.

This would be the same at 2nd or 3rd.

6 feet from first? Protected between 1st and 2nd. That's 2 steps. You don't think the fielder's existence right at that spot altered the path the runner would have taken between 1st and 2nd?

Unless there was a play on her and she was clearly intending to run straight through first base, you've got to protect her between 1st and 2nd.

Six feet (two steps) prior to 1B seems to be between HP and 1B not between 1B and 2B. The BR could be protected to 2B but not under the "between the bases" part of OBS. The umpire would have to judge the BR had a chance to make 2B sans the OBS prior to 1B would s/he not?

This was my thought process, exactly. I'm interested in hearing more opinions one way or the other.

Here is a similar situation, how would you guys rule?

B1 squares to bunt, causing F3 to charge up the line well in front of 1st base. B1 then pulls the bat back and slaps one into CF which is fielded by F8, who is playing in. F8 throws to F5 covering second. As B1 runs up the line, she is bumped into by F3 and knocked off stride. She then continues running and runs into an easy out (by 20 ft) at 2nd base.

The obstruction clearly happened between home and first base. The runner made a choice to attempt to get second base when she had no realistic chance to do so.

I am not protecting a runner who makes a poor decision, or rewarding a coach for making a poor decision to send the runner.

For a change, the easiest answer is for NSA rule set.
Dead ball immediately with obstruction on BR prior to reaching first base.
Perhaps in place for this vary reason....

I agree with your thought process.