You're a bit misinformed, he was not duke of Holstein, but Duke of Holstein-Gottorp. The actual possessions he directly ruled are rather limited, as the Holstein-Gottorp line lost the majority of its territories in the Great Northern War, which was located in Schleswig, that Peter wanted to gain back.
Orange = Kingdom of Denmark.
Dark Orange = Gottorps lost possesions to Denmark in Schleswig, and the County of Rantzau's return to Denmark in Holstein.
Light Yellow = jointly ruled areas between Denmark and the Gottorp line.
Teal/blue/purple = Gottorp possessions.
So what you're saying is Peter is going after Schleswig, but the rest of Holstein he has no interest in ? Because To be honest if he is just going after Schleswig (andt his that all parts or just the previous Gottorp parts?) then sure, he is not gonna gain access to any impressive North Sea Habours.
What I'm saying is not important. What important is what Peter III was saying. The official position was that the war is about Schlezwig. He was seemingly OK with whatever he had in Holstein and what he was getting out of it: few hundred of his "Holstinian" troops with which he was playing in Oranienbaum. Also, he had a right to award Holstinian order of St. Anne which was retained by his son, Paul, even after Catherine settled the "holstinian issue" with Denmark.
Why did he decide to be so concerned about Schlezwig nobody (AFAIK) can tell for sure. Was the whole thing 100% rational? I don't think so. But being rational was not a "must" for the Russian monarchs and Peter just got Russia out of a much greater mess, 7YW, rationale for the Russian participation in which was even more questionable.
The whole issue of the North Sea harbors, under ANY imaginable scenario, is absolutely irrelevant: for the Russian Baltic fleet of that time just getting out of the Baltic Sea (with Denmark being friendly) was a major task involving noticeable losses. Usually, it was maneuvering inside the Gulf of Finland and that was it. It took Catherine's Ottoman Wars to start developing Russian navy but for her policies the Northern Sea and its ports was not important at all: British ports had been much more useful on many accounts.
But if he is going for it all, which seems much more likely, he will be going for Altona, Denmarks largest North sea habour. You're still ignoring that Denmark remained friendly with Britain during this period. It's all fine and dandy saying that they are now friendly with Russia because of Peter's stunt, but ignoring the other side of the diplomatic scale seems rather odd.
No, Britain was not friendly with Russia "because of Peter's stunt". It was friendly with Russia even during the 7YW when they were on the different sides but not at war with each other. And the reason for this friendliness was quite material: existing trade relations had been
extremely important for both countries (Britain needed Russian timber, iron, hemp and grain and Russia needed revenues).
Peter's intention to go to war over Schlezwig did not result in an immediate military action so the Brits would have enough time for the diplomatic interference if they wanted. Neutral position would suit them just fine: their trade on the Baltic is not impacted (I'll repeat one more time: practically all Baltic trade with Russia had been conducted by the British merchant ships so Denmark could do nothing to it without serious consequences) and they could offer a mediation at any time.
So all of it? considering the Danish king rules all of Schleswig during this period. Or just as the previous parts of Schleswig that the Gottorp dukes ruled?
And Peter is simply okay with being joint ruler of Holstein, having the least to say in that duchy? It is after all the richer part of Schleswig-Holstein?
Yes, it seems that your "program minimum" amounts to pretty much everything Peter was looking for: "his" part of Schlezwig. As for Holstein, he never ruled it to start with, being brought to Russia as a boy. AFAIK, when he grew up (still heir presumptive of Russia) he did not interfere in the Duchy's administration in any serious way beyond, as I said, requesting a detachment of soldiers.
So, what we are talking about is more or less much ado about nothing. The armies would do some ritual dancing. In the worst, for Denmark, case scenario, Rumiantsev forces the Danish force to get on the open and gains a victory: staying all the time in a fortified position was not the best strategy as had been demonstrated at Pirna, especially when the opponent has enough numbers to keep you in that position AND to do something nasty to your countryside (*). But these details would not really matter too much because very little is at stake and none of the sides suffers from excessive bloodthirstiness toward opponent.
Soon after the whole mess starts, it is almost definitely going to be some diplomatic (either direct or with the Brits as intermediary) action with Peter ending up getting his favorite cookie (few pieces of land which he is never going to visit). There are celebrations in St-Petersburg, Rumiantsev is getting his fieldmarshal's baton and the Danes are issuing a sight of relief (and perhaps celebrating as well because they did not lose anything worth mentioning).
In OTL Catherine II settled the whole issue (acting on behalf of her son who inherited father's title and claim) within few years after Peter's murder by ceding the territory and claim (except rights to the order of St. Anne) to Denmark in exchange to Oldenburg which she gave to her uncle: within her political framework these territories and messy claims had been just impediments. She was playing ultra-Russian card which means that aggression is going to happen in a completely different place and it does not make sense to have unresolved complications over the issues which do not matter.
However, Peter was a different personality. Definition "crazy syphilitic" (if I remember exact words) actually describes his grandfather, Peter I. PR campaign skillfully launched by Catherine after his assassination created an image of insane person but it works only if one keeps ignoring the fact that during his short reign he managed to do a number of very important things. Some of the major ones:
(a) Got Russia out of the big expensive cabinet war. With all brouhaha about the "betrayal" Catherine did not change anything in his peace treaty with Old Fritz and maintained very friendly relations with Prussia. Intended war with Denmark was considered by the contemporaries as a small potato. Except for the Gurads who did not want to leave St-Petersburg and preferred to change a ruler.
(b) Decree of the "freedom of nobility" - abolished life long mandatory military service for the Russian nobility allowing them to start paying more attention to their estates with a resulting general improvement in the Russian economy. Everybody was quite ecstatic: Peter was offered a golden statue to commemorate this decree but modestly declined.
(c) Abolished Political Secret Police (which his wife immediately restored).
(d) Abolished physical punishments (flogging, cutting the tongue, etc.) for the nobility.
(e) Was planning to announce equality of the religions. Now, THIS raised a lot of hackles but can we say that this was a bad thing?
(f) Tried to introduce at least some military discipline and training among the Guards. This was a killer (for him) but done by his successors.
Of course, he had some peculiarities (seems to be in a family) and Sclezwig thingy was one of them. How would it play in a long run we can't tell. Probably it is safe to assume that, if he survived, this mostly titular, "possession" would last during his life time to be inherited by his son. Would he be able to make some practical use out of it? I don't see how. Surely, he was NOT going to incorporate it into the Russian empire, which means continuation of the business as usual. His son is one more grey area: we know what he was in OTL but this was a byproduct of the decades of a neglect and even open hostility from his mother so ATL Paul could have a seriously different personality. Anyway, the options are the same: either a titular Duke or a land swap. Chances for the swap will keep growing with each next emperor.
The only potentially interesting situation arises in a highly unlikely case when the whole thing keeps going on until the mid-XIX when Russian emperor is still a titular Duke of <whatever & whatever> and the 1st Schlezwig War is on a horizon.
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(*) One of Pratchett's "rules of war": if your opponent occupies a very strong position, make sure that he keeps occupying it.
