If I were doing this new series, here's the route I'd go as far as explaining future Belmonts.....and yes, Alucard himself.  
"Gabe has a half brother back with his real Cronqvist family who finds out about him in his formative years.  This brother (who's name is Adrian) becomes furious when he learns that his father abandoned his own bastard child to some random monastery....and goes looking for Gabe.  Adrian finds the remnants of the Order and the monks that raised Gabriel.  They tell him of Gabriel's legacy...how he became a great holy warrior and "died" destroying the Lords of Shadow for the sake of all humanity.  Adrian, swollen with pride for his brother's accomplishments (and as an act of spite for his father) takes the name Belmont to honor his fallen hero of a brother.  He takes up arms in the name of serving the church, and helps to start rebuilding the Order of Light.  
Years later Adrian has married and has a son (or sons).  He meets a long lost high ranking member of the Order who tells him of the vampiric presence that still haunts the hills about the town of Wygol.  This mysterious figure also tells him that the castle overlooking this town is where is brother Gabriel fell during his final battle against the Lords of Shadow.  Adrian, naturally, volunteers to venture to this dark castle to try and root out all evil that still roams there........and to perhaps discover the resting place of his lost brother.
Adrian journeys to Wygol, and fights his way to the spire of the Dark Keep.  When he arrives, he meets the keep's dark lord.....who only reveals himself as "Dracul".  Dracul recognises Adrian's garb, and asks him his name.  When Adrian tells him his name is Belmont......Dracul becomes enraged.  Convinced that the Order sent this man (labeled with his old surname as an  insult) to assassinate him, he attacks Adrian.  After a brief fight, Drac is impressed with the warrior's fighting prowess...and the two begin to jaw for a bit.  
i liked it all the way up to here. VERY good way of reintroducing Belmonts, and Castlevania, to the best of my knowledge, has never had a brotherly conflict like this. (or has it?)
now, afterward, I have a problem.
Drac is just revealing his true identity to his guest (much to Adrian's horror and disbelief) when one of Drac's vampire lieutenants enters and tries to attack Adrian.  Drac intervenes.....hurling the Combat cross (previously hidden beneath a cloak near Drac's throne) at the greater vampire, impaling him and sending him through a window....where he crashes upon the mountain and tumbles down to the outskirts of Wygol.
That is assuming Gabriel can even weild it anymore, now being one of the lot who it was made to kill. Not to mention that leaves the new Belmont protagonist without a whip, something which is completely unacceptable. It would make sense that he would no longer be able to wield it, and would likely discard it, along with God and the brotherhood, taking on the name of Dracul. Who knows, maybe Adrian would encounter A mysterious old beggar who gives it to him near the beginning of his journey, (rinaldo)  or it is simply taken back to the brotherhood and he decides to take up the weapon of his brother. 
 Instead, Drac bites him (turning him).
Doesnt work that way. He would have to receive Gabriel's blood to become a vampire. (IOW, how it originally was with vampires, and SHOULD be) And for that matter, why would he turn his ENEMY into an immortal undead creature on the same power level as himself? he would be more likely to kill him and drain his blood, not make him his kinsman.
Also, if Alucard is not Dracula's son, then there is just no meaning to it. ESPECIALLY in Castlevania. Even in the original usage of the name, it was Dracula's son. (the Lon Chaney movie, the son of Dracula) thats where it comes from. Now if you want to name the result Joachim, then theres something we can work with. but Alucard must always be Dracula's son.
heres how i see it. it goes down as you described, brother and all that, but lets include a few more story relevant bosses. And he slowly learns of Gabriel's journey through them. In fact, you mentioned a long lost order member? Well, who better than Zobek? he fooled Gabriel, and even the brotherhood scrolls seemed to indicate he was still around, (they name him as one of the longest serving order members) Zobek meets with the Belmont, and plays a similar role as he did with Gabriel, misleading him and feeding him false information. Maybe for some ulterior motive... Who knows. perhaps he wants him to suffer the same fall as Gabriel. And eventually, he would reveal himself as who he really is, and you would get to finally fight Zobek in his true form, as lord of the Necromancers, AKA death. 
you fight Gabriel, who by now, is fed up with God, (maybe with humanity too?) and declares that he will kill any who stand in his way. you fight, and upon his defeat, he proclaims he will get his revenge. And in turn, his brother proclaims he and his family will hunt him down until he dies, using His own holy  weapon against him. only issue I can see, is how we can bypass his dying for good as a boss. perhaps since Zobek is lord of the dead,Zobek has made sure that Gabriel, when dead, will never actually go to the realm of the dead, but be brought back. Maybe the castle and it's creature of chaos-ness can factor into all this too.
what do you think?