If I read the OP correctly, the BR/R did not leave the field; but was tagged by a live ball while off the base and not returning directly. That is an out, if it is treated as an appeal.
We have discussed recognizing such a play as an appeal several times; which is the deciding factor in this case.
I would say it is; if the fielder(s) acted as if they were trying to catch the BR/R in a running violation.
If the fielder just tagged the BR/R haphazardly with no apparent intent; then it's back to a question of what constitutes an appeal.
BTW, the OP is very well done, clear and lacking useless embellishment.
