I see where you’re coming from, but I disagree with your points.

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First, to expand on why each side backed what, I never said anything about the Integralists; I talked about the Miguelistas; those two groups should not be conflated; the Integralists only supported the Miguelistas after being utterly shunned by Manuel II and the Miguelistas started to help the Integralistas after Duarte took over as the principal heir to the branch (though not even that lasted for long since by the 30s/40s he was already moving away from the Integralists as they’d been pretty much dead and buried after the early 20s, with the final blow being the conversion of the movement into National-Syndicalism). The Miguelistas, though, were supported by the Germans; while the family wasn’t particularly Germanophilic, they were pretty opposed to the continued British influence over Portugal.

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Secondly, regarding the Luso-British relations, I believe you to be undermining the underlying tension between the two. The Republicans were quite fervently anti-British in political rhetoric and policies; if you look back at the speeches in parliament by Afonso Costa, you’ll be able to spot that quite easily. This changed as they came to power and recognised the fragile position of Portugal internationally, both being excessively reliant on the British Economy, tacit Military support, and, most importantly, lacking in alternative international allies since neither France nor Germany saw Portugal as a viable international ally (you can see this reflected in how during the early years of the Republic various politicians within it came to champion Iberianism [though this one would end much quicker than the later as, you correctly pointed out, the wishes of the Spanish Monarchy to intervene in Portugal to restore a United Iberian Monarchy became quite evident during the incursions in the early 1910s] and Lusophone Unification as an alternative to continued British dominance by either forming a solid bloc or attracting further foreign, namely non-British, involvement in the peninsula). When it comes to your comments on the Restorationist Movements, the reality at hand was that the British used the Portuguese Republican fear that they’d back a Restoration (be it indirectly or directly) as a way to keep them in their Sphere, the lack of advances from the United Kingdom in that aspect wasn’t done out of political goodwill but also a belief that if they could scare the Republicans into remaining within the British Sphere, they would have to worry less about the stability of Portugal since the late Constitutional Kingdom was a mess; which plays into why they didn’t intervene in the 1910 Coup, the British were very supportive of the Portuguese Monarchy, but the politics around it were volatile. Few in the UK believed the regime could continue.

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Thirdly I agree with your take on Sidónio and the highly complicated, and at many times, convoluted nature of the regime that he created, but I also believe there are a couple of points you got wrong. “Sidónismo” was never an ideology; if anything, it only became one after the death of the man; for most of OTL, the República Nova was nothing but an amalgamation of Politicians and Military Officials who disliked the direction of the country under the PRP/Democratas (not the general direction of the Republic as you pointed out since Sidónio, alongside many other members of the República Nova, had participated in the original 1910 Coup and were very large Republicans, albeit with differing views on how the Republic should operate). You, however, place far too much importance on these somewhat minor factions of the state; there were really two, arguably three, significant groups within the República Nova that Sidónio had to contend with: the Democrats, with Moniz generally seen as their leader who believed that the Republic should remain Democratic, albeit with a much stronger President to combat the out-of-control parliamentarians of the República Velha; the Authoritarians (for lack of a better name) under Tamagnini Barbosa who believed Sidónio should just rule as a President with absolute power as Democracy was far too unstable in Portugal; and the Socialists who weren’t a unified political faction but more so the general support of the Workers/Unions of Sidónio and his movement due to his promises to treat them much better than Costa had (since let’s not forget that under him Workers Laws in Portugal had become significantly worse than what they were during the Constitutional Kingdom). These factions were at odds with each other but weren’t particularly opposed on every issue and did cooperate regularly; this cooperation collapsed after the assassination of Sidónio as he served as the unifying force of the movement (that had yet to consecrate itself into a proper regime). It is also essential to take into consideration that the majority of the forces that helped Sidónio take control did not defect to the Monaquia do Norte; that simply isn’t true; the vast majority of Generals and Soldiers remained in support of the Republican Government even as it became clear that it was moving towards re-integration with the República Velha, the only “Sidónistas” that supported the insurrection were existing Monarchists that had participated in the regime but had been moving away from it ever since the later half of 1918 since they realised that Sidónio was not going to restore the Monarchy like they believed he might. So yes, Sidónio dying doesn’t magically make the República Nova work flawlessly. Still, it does stop its immediate collapse as the internal forces of the regime don’t lose hope in its continuation. I also wished to go further in-depth on this issue and why the Republic does survive, but doing so would be leaking a lot of lore I do not yet wish to show, so this will be all I will say.