Disclaimer: Note that I will put "good" and "evil" in quotes almost exclusively, due to the dichotomous nature of the terms as I'm discussing and referring to them. Bear that in mind.
General personal thoughts:
-Demon King is a title and rank, not a singular one-time being.
-Christian God plays a decently-sized role in CV lore, but is clearly not defined as the sole deity in existence in the universe. Therefore, it's not particularly prudent to take many major Biblical stories and assume they're true in the world of CV. Hell, I could even make a case that, since a lot of Christianity descends from Jewish roots, that as per the Jewish faith under Jehova's rule there IS no Hell as Christianity defines it. No Hell, no single angel-that-fell Satan.
-Pretty much all references to Satan as the singular being we all know him as are the fault of mistranslations.
Personal headcanon:
-The title of Demon King is a divine role, much like Death's. By this, I draw upon the dichotomy that good cannot exist without evil as a reference of the inverse, much how light cannot exist with darkness. To this end, I consider the Demon Realm/Chaos/whatever one wishes to call it to be a very tangible location much like the Heavens of the many deities we see in the lore. Such a realm would need a ruler, and that ruler holds the title of Demon King. Perhaps there was a Satan or Lucifer or Samael holding power there are one point, prior to Dracula's claiming of the rank. As we see in Dawn, it is a role that must be fulfilled for the realm to exist properly, which I feel lends credence to the notion that it is a divine realm and role that exists as a counterbalance to the realms and roles of what is considered "good." Should such a Satan or Devil have existed within the lore and universe, I doubt very highly that he was in any way commanding or otherwise influencing Dracula.
-Death's relationship to Dracula, or more specifically the Crimson Stone, comes from a very simple concept--the Crimson Stone grants fabulous dark powers, with its owners pretty much guaranteed to perform "evil" acts with it. Such acts would naturally result in people dying in large numbers, something I'm sure Death has a very high investment in. Therefore, I consider his attachment to Dracula to be predominantly one of "okay, whoever holds the Stone is gonna probably kill a shitload of people, and this Dracula dude's been doing a damn fine job of that for centuries, so if I hang with him I'll have lots of work!" Their implied friendship/acquaintance I'm sure simply developed as a natural byproduct of the time they've spent in collusion. Pacts can be made with Death without the Stone, but in those cases it simply becomes a matter of Death siding with the highest bidder--canonically, Walter seems to have some kind of deal with him, but Death abandons it the instant a more promising partner/master shows up, Mathias.
-The Crimson Stone is essentially a form of the Philosopher's Stone that grants the legendary powers of it in a way humans wouldn't even consider as valid due to its "evil" connotations (I am paraphrasing the ever-loving SHIT out of this headcanon, as I can very literally explain on it in borderline-disgusting detail for HOURS on end--Plot can confirm this). Consider that the historical Stone is always described as granting amazing powers and immortality. Well, what does vampirism do if not exactly those things? But since vampires are considered evil, then human alchemists such as Rinaldo would look at the effects of the Crimson Stone and go "no, that's just a bastardized failure, there's no WAY that thing can be a true Stone," despite it granting a form of the exact abilities humans claim it to possess. Since I assume this to be the case, then that makes the Crimson Stone something that requires ingredients and a recipe, therefore it is something that must be WANTED to be made. Ergo, someone WANTS to make such a stone, therefore WANTING what it gives, therefore see above about Death and why I feel his attachment to the Stone's owner to be as it is.
As for how all these tie together to relate to the OP topic:
-The Devil as we know him may or may not have ever truly existed in Castlevania. There doesn't seem to be much concrete canon evidence declaring as such, with "Devils" being relegated to simple powerful demons bearing the name, rather than there being a single "true" Devil. If he had existed, it doesn't seem to be the case anymore, or at the very least he does not exist in his seat of usual power.
-The title of Demon King appears very strongly to be one of rank and power, rather than a singular being like Satan. Since it's such a title and rank, it can change hands and be usurped by stronger entities. It also appears that such a leader is required for the corresponding Realm to be whole and functional, thus further signifying that there doesn't seem to be a universal all-powerful evil entity like Satan in control of it at all times.
-Since the title of Demon King seems to be a temporary title that can be usurped by a more powerful willing being, I think it's fair to assume that Dracula has all but filled and maintained that role for a millennium with no candidates stronger than him showing up to take it from him. Since that's true, I would imagine that, if a Satan still existed, he would've been trying to reclaim that power in Dracula's many naptimes. Since this doesn't appear to be the case, I believe it can be reasonably inferred that regardless of whether or not Satan as we know him in Christianity ever even existed and held power in CV lore, that mantle is now held by Dracula alone. Nothing else "evil" has ever successfully usurped him or his seat as Demon King, the Belmonts seem to have been selected as God's Chosen warriors to combat this force of unrivaled evil and darkness, Dracula bears explicit weaknesses to Church and other holy symbols and effigies (arguably beyond even the traditional threshold of vampire weaknesses, as Dracula has transcended far beyond an ordinary vampire at this point, and is more a dark force of nature due to his heavy involvement with Chaos to such a point that calling him just a vampire would almost be an insult) that aren't necessarily even Christian in nature. If he displays vulnerabilities to holy elements from faiths other than Christianity, it would suggest that his alignment exceeds simply that of the Christian Devil and puts him more on par with that of a universally wicked force that opposes ALL that is divine and "good."
-In sum, if there were a role which we could consider that of Satan, then Dracula's taken it over and had it for a very long time. As far as the CV universe and lore goes, Dracula more or less IS the Devil of the world. He obtains the necessary title of Demon King, holds it unchallenged for a thousand years, and when he's destroyed for good nothing seems to jump in and take his place for almost four decades--suggesting that he either did a damn fine job of clearing the playing field, enslaved any competitors to his will and slew those who would not comply (or in the case of the only KNOWN would-be usurper of the mantle of Demon King/Dark Lord, Galamoth, confined them within the Castle and the heroes killed them), or simply left such a mark on the other denizens of the Demon Realm that even after his "true" death in 1999, his residual power in the DLCs and Soma was enough to strike fear into the Realm's hearts as to just another delayed resurrection, and stay them from trying to take over.
So the TL;DR of all this is that, as far as the lore and canon seem to suggest, Dracula is basically the Devil in the CV universe now, regardless of whether or not the Christian one existed prior to him.