I'm thinking that that was an astute bit of typically Hohenzollern politics. The French and Spanish, Bavarians and English and Dutch wewre not going to come to Brandenburg and lay siege to its cities. The Swedes or Poles just might have. And there is no guarantee of backing a winner.
Oh, as said, it was good foreign politics IOTL. But with the two wars happening near-simultanous, and Brandenburg in the middle between them, it comes as kinda a natural question if you think about the period, and Brandenburg in special.
And, btw, no, we cant refer to it as Prussia:
Prussia was raised to Kingdom when Habsburg promised to accept that to win Brandenburg over for the War of the Spanish Sucession. It then became the most senior territory. Brandenburg not particpating in the War of Spanish Sucession means Prussia wont get raised to kingdom, so its at best Brandenburg-Prussia, but thats too long for me...
Now, obviously, Brandenburg has two options in the war: Side with the Alliance, or side with Sweden. IOTL, they made soem agreements withS weden, and tried to support a guarantee of neutrality of Swedish Pommerania, and thats as far as they leaned into the war. More pro-swedish than pro-Alliance, but not enough to deduce anything. Brandenburg could indeed have joined both sides.
Indeed, it was invited at first by Denmark, Saxony and Russia to join their war against Sweden. A natural offer, as it was well known Brandenburf desired Swediish Pommerania (and had a very good claim to it, too). Brandenburg could have joined the coalition right frm the start. The question is now: Does the Brandenburger army make any difference? Prussia,Brandenburg and Pommerania would be more or less on Charles' path anyways, solittle would change in that. The question is if the Brandenburgers can turn the tide... and actually, that could go so much both ways we now have three possibilities. (It should be noted August of Poland and Saxony was in the end desperate enough to promise Royal Prussia to Brandenburg if only itwould intervene, so if Brandenburg plays it well enough it can gain from both sides).
OR Brandenburg could side with Sweden. Maybe Frederick is unsure in the beginning and waits, and then sees the incredible sucesses of teh Swedish Army in Poland, and decides for joining up in beating on it. Of course such a timing is incompatible with matteops scenario but I really cant see Brandenburg joining earlier, when everybody thought Sweden would lose. In which case Charles indeed probably still sets off to fight the Russians, with the predictable results, and Brandenburg is left alone - then again, it would only square off against a recently devasted Saxony, and it can do that. (It would probably would receive some Saxon territories, too)
Of course, Brandenburg can also flip sides, but that would require it to start on the coalitions side (really in the scenario of the directly above paragraph, why would the coalition powers eager for revenge and BRandenburg spoils accept a flip?). How that works out depends again on how well Brandenburg can pull it off... either great disaster or great victory.
So, lets say, four possibilities:
1) Brandenburg joins the coalition early, but it still loses nontheless. Brandenburg gets devasted, and may have to cede Eats Pommerania, but takes it back (maybe together with West Pomemrania) when Charles get lost in Russia - but in the end effetc it still makes a loss due to the devastation.
2) Brandenburg joins the coalition early, which causes it to win against Charles in Poland. Brandenburg gains Swedish Pommerania, and as said if it plays it well enoughand Charles remains a threat, maybe Royal Prussia a swell.
3) Brandenburg joins the coalition late, maybe in return for August promises of Royal Prussia. See above.
4) Brandenburg joins the coalition, but then backstabs them, and maybe back again, etc. It ulls it off sucessful, and gains various territories from both sides depending on the timing and other factors.
5) Brandenburg flip-flops, but fails horribly at that. See scenario 7.
6) Brandenburg joins Sweden, and stays at the Swedish side. Charle sget lost in Russia, Brandneburg stays behind - but as everybody is warweary, nothing happens.
7) Brandenburg joins Sweden, and stays at the Swedish side. Charles get lost in Russia, Brandneburg stays behind. Russia for some reason is vengeful enough to come knocking, together with Saxony and Denmark. The Hohenzollern domains are torn apart.