Interesting thoughts. The involvement of Brandenburg and Sweden, I think, is the key issue. Although they seem like plausible Russian partners, neither is in a very good strategic or political situation at the moment.
As I recall, Peter III did try to interest Sweden in his war and offered them Norway, but the Swedes were not in a position to accept. The "Pomeranian War" with Prussia had just come to a ruinous end, leaving Sweden with nothing to show for a huge amount of money spent and thousands of lives lost. The government was broke and facing a crisis of confidence at home. TTL's Pomeranian war was shorter, presumably less costly, and was actually "successful" in a strict sense (the Swedes performed rather poorly but managed to get Usedom and Wollin in the peace treaty), so one could argue that they are in a better position to fight another war. This actually makes war less likely, however, because a "success" in Pomerania will presumably bolster the Hats party then in power, which was pro-French and anti-Russian. Under such leadership, an alliance with Peter intended to expand Russian power in the Baltic seems very unlikely, even if Norway is offered as an inducement.
With Brandenburg, the issue is how long the postwar recovery is going to be. While the electorate remains in the top tier of German states, its core territories have been ravaged, its army is presumably in a shambles, and the state itself is under a minority regency. A Russo-Brandenburger defensive alliance is an eminently sensible idea, but it may be some years before Brandenburg is really prepared to launch another major offensive war. Your scenario seems more plausible a bit further in the future, after Brandenburg recovers its strength, but I'm not sure if Peter has that kind of patience. Additionally, we're making an assumption here that Brandenburg's leadership will continue to be ambitious and aggressive. Frederick's campaign of aggrandizement came to a disastrous and ignoble end ITTL, and his young successor might not be too eager to follow in his footsteps with a campaign of naked conquest against a neighbor. Not every Hohenzollern is Frederick, after all.
As a side note, I think Britain is very unlikely to offer any help to Denmark beyond diplomatic support. Britain's foreign policy is in a severe crisis right now, as with Austria's defection and Prussia's defeat they are left without any significant continental ally. Russia is their great hope for a new friend on the continent (as it was historically), and I suspect the British are unlikely to do anything that might damage prospects for an alliance. Giving material aid to Russia's enemy definitely qualifies.
I think you bring up ssome good arguments.
I agree that if the Swedes are smart they stay out, but Sweden did a lot of stupid things in this period. The potentiak rewards are large, but the the same time it comes at a very high price of having Russia on their western borde.
As for Brandenburg, the potential reward are large for Brandenburg, but it comes at a price. Brandenburg have less than zero claim to any territory controlled by the Danes, so it would send a signal of the new Elector of Brandenburg being a untrustworthy rogue like his uncle. If Russia lose, Brandenburg may not lose any territory (through that's not a entire safe bet, Denmark may take East Frisia and maybe some of the western exclaves), but Brandenburg have ensured a hostile Denmark and Brandenburg without Prussia and Silesia is a smaller power than Denmark, and a much safer target of Danish revenge than Russia.
As for Peter III, I think his survival as Czar would be interesting. The war with Denmark is less dumb than OTL "war", and while I don't think he will win (at least not without Brandenburg joining him) as Danish logistic gives them an edge, he could very well survive and maybe even learn from it. Honestly I could see him starting the war, winning and/or losing a few battles, only to have to deal with a coup attempt at home and maybe surviving it. This force him to make a fast and humilating peace with Denmark, where Denmark buy Holstein-Gottorp (so that it look less humilating) and he plan for an later reconquest, when he have established control at home (which end up taking most of the rest of his life).
Denmark here is forced out of its position of neutrality simply because of fear of Russia, at the same time Crown Prince Christian VII of Denmark are born after the POD which mean his madness is butterflied away. The Hessian princes still end up in Denmark (the reason for them ending up in Denmark is unlikley to chane). The lack of madness again means a later marriage marriage for the crown prince (his sister likely marry much as they did in OTL), the hostility to Russia likely mean no British marriage for the prince, a interesting alternative marriage would instead be Princess Sophie Albertina of Sweden (or technical her hypothetical doppelganger who share the same name), which would bind Denmark and Sweden closer, which Denmark would have interest in with a hostile Russia.