Coverage at 1b on walk

[CecilOne]

(PU = plate, BU = bases)
1) A violation of the RCR is not a specific responsibility for anyone, it belongs to whoever is responsible for that runner.
2) Just a stop is not a violation unless the runner stays at that spot beyond your sense of "immediate" in "immediately" moving toward one base or the other.
3) Give the runner the benefit of the doubt.
4) In you scenario, R1 ran on the pitch and if the BU went with R1, the BU can't be watching for RCR at 1st until R1 stops. Unless of course, both umpires realize that the lead runner is PU resp. and the trail runner is BU resp.
5) The PU should not be trailing a BR to 1st with a runner on 2nd or 3rd.