In MM, the other Harry will attempt to break out Demonreach inmates because he will be attempting to save someone [ e.g.: Susan or a lover from the alternatvie reality. ]. I mean: It's Harry. By now we kinda figured out his personality and we can see somewhat see how life is shaping him... and we can all go back to book 1-4 to re-read, if necessary.
If the alternate Harry made other choices, different lovers, different something... he's a different Harry, but he's still has a remainder of "young Harry" in him [ otherwise the parallel angle won't make much sense ]. So however and why ever they are going to meet, I think they will try to cooperate whenever they are not working directly against each other.
But at some point the Alternate Harry will say that he's not the Warden of Demonreach, doesn't have Soulfire and wouldn't want to attempt to do the same in his universe, but Harry already has the access and Alternate Harry _really_, _really_, _really_ needs to get something/someone from there to help him save someone else.
Harry will naturally try to explain what kind of monsters are locked up under Demonreach and why and Alternate Harry will be either more reckless or be more sure of himself and sneak onto the island meeting a very confused Demonreach, who sees his Warden, is talking to him, can sense him every particular manner and he knows it's his Warden, but... doesn't feel him. There is no connection. And the Alternate Harry, being a bit of a bullshitter, will make something up and insist on retrieving something from underneath... and somehow [ I can't imagine how ] "persuades" Demonreach...
As Harry arrives, it's too late by a few seconds - some [ relatively low lvl ] monster has been released. Apocalypse lvl 1 starts! ... Harry, Alternate Harry and allies somehow fight it off/kill it/imprison it again. Alternate Harry either dies or someone from his close allies dies...
Add White Council being wary because interdimensional travel, Mab, because there is another Dresden, who will probably try to help Harry get out from the service... and anyone else who tried to recruit Dresden.
I mean: Not even Merlin jumped between dimensions! Did he...?