One big problem they're likely to run into though is that, by placing the political capital so far south and shifting the national focus to the Black Sea/Cauauses/Steppes as a consquence, the naturally more populated and wealthy/productive Russian heartland and north is liable to develop an alternative "center of gravity" that will weakn centeral Czarist authority.
The whole schema (no matter which of the candidate cities is chosen) involves, as a prerequisite, conquest of the Khanate of Crimea, dealing with the Nogai Horde, removing the Ottomans from the fortresses on the Black Sea coast (depending upon scenario, Kerch, Ochakov, etc.) and getting the Ottomans to agree on a free traffic through the Straits. Implementation of that program means availability of the fertile lands to the North of the Black Sea (region controlled by the Crimea). Which, in turn, means that the area can be safely populated and that the former border area of the Left Bank Ukraine is safe. Which means that the area can be populated reasonably fast and provide a considerable volume of the agricultural product. Production from the Central Russia could be reasonably easily shipped to the newly open Black Sea ports both by land and by taking advantage of the existing river routes (with an obvious but not necessarily successful attempt to build Volga-Don canal).
While it can be said that St-Petersburg was closer (as crow flies) to the Central Russia (actually, I'm not so sure), it did not necessarily had a more convenient communication with it than one which would be provided by Volga-Don route: quite a few economically important cities already were on Volga, including Nizny Novgorod, and the river routes had been more reliable than the land ones. In OTL St-Petersburg created "shift of the gravity" without any noticeable weakening of Tsar's authority.
IOTL, one of the reasons Russian managed to be so powerful was they were able to bring establish an absolutist state and so concentrate the full resources of their state on iniatives as well as easily propagate reform (as opposed to the, say, the Ottomans who's level of decenteralization was serving as a hinderance to similar reform efforts in the early 1800's). ITTL I could very easily see the Novgorod/Muscovy region resisting any attempt to keep a tight hand on the reigns from the distant court, especially considering how powerful "Old Russia" culture was in the regions surronding Moscow in contrast to the reform-minded, Europeanesque culture increasingly being pushed by the court under Peter.
Rather fanciful on more than one account.
There was no "Moscow - Novgorod region": thanks to the consistent efforts of the rulers starting from Ivan III Novgorod became just a provincial city (from 1727 a capital of the gubernia) much less important than Nizhny Novgorod which by the XIX century grew into the trade capital of the Russian Empire. And don't forget other important cities in the area, like Astrakhan. Then, pretty much all that "Old Russia" stuff was pretty much non-existent and western-style costumes together with the smoking and excessive, even by the Russian standards, drinking (which was more or less what "westernization" at Peter's court amount to) had been adopted well before creation of St-Petersburg.
This hampers Russian power in the medium-long term, and runs the very real risk of the north even trying to break away (or, at least, the locals developing an alternative enough mindset to try to seize control)
The "North" as far as Novgorod is involved would not go anywhere (the last time its inhabitants had their own mindset was during the reign of Ivan IV and when being occupied by Swedes during the Time of Troubles the area did not develop any clearly visible "western" tendencies) and trade via Swedish-held Narva, Riga and Revel would continue.
