I would think that he will be on the active roster as well.
There are a number of other teams that have this same situation. Ewers in Miami, Leonard in Indy to name a couple. Rookies who have impressed but who are probably QB3. The decision is a tough one as it would be ideal to not use a roster spot on someone that is very unlikely to see the field this season, especially when that extra lineman is so valuable to a team during a game.
The thing that makes Brosmer's situation so different than most of the other teams with this decision is that he a backup to a guy who the young future of your team. For example, Miami can easily think that Ewers should be developed to be their future starter over an extremely injury prone Tua or Leonard over Daniel Jones who has failed in the past. Whereas the Vikings could question, what are they developing Brosmer for? To be traded to another team? To put pressure on McCarthy? But things change, injuries happen, so who knows.