No, that’s not your responsibility. As the COR, your responsibility is to make sure your chartered organization is supporting the troop, though. Also, the church owes both units a reasonable accommodation – on the one hand, they’re chartering one troop (so they should be actively supporting it as part of their youth development programming – that’s what the charter agreement says) and on the other hand, via benign neglect, they’ve allowed the other organization free access to meeting spaces for years and years.
I’m really surprised that your church has been sanguine about another chartered organization running a troop on their premises… There are liability and legal issues in play there that could put the church at financial risk.
Were I in your position (and to be honest up front, this wouldn’t make me any friends), I’d push the issue with the church and ask the pastor to tell the other unit that it’s time to find a new meeting place, since the church now charters a Troop as part of their youth ministry.