It does make sense that the rule is by appearance not innings but I wasn't sure. They are just lucky Brogdon is re-emerging as a late option, especially as Familia doesn't seem to be.
Bohm is in a slump while the team is 8-0, so just let him play through it probably, at least until they have other players. They could give Vierling a game there but probably better to just let him face a lefty in CF.