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.  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.
Adrian...ignoring Drac's momentary charity....tries to seize the opportunity to attack his quarry one more time.  Drac's patience runs out, and quickly subdues him.  Adrian taunts Drac to finish him, but the vampire lord tells him that he has a far worse fate in mind.  Instead, Drac bites him (turning him).
In the aftermath, Adrian flees to the bowels of the Dark Keep.  After his "becoming", he remains hidden below.....unwilling to venture back out amongst man (where he knows his new nature is likely to cause him to kill innocents).....but unable to end himself (lest his God damn his soul).  He swears vengeance against his maker....eventually taking the moniker "Alucard" as a spiteful jest against his creator.
Meanwhile, the townsfolk of Wygol recover the Combat Cross, and assume that the brave Belmont warrior gave his life to destroy one of the Keep's vampire lords.  The Cross is eventually brought back to the Order......where it would await the hands of one of Adrian's young sons in times to come."
Yes, I changed Alucard from being Drac's son to being his brother, but I think it fits just as well.  In fact, it provides a much better motivation for Alucard's cause against his "maker" than him just rebeling against "dear old dad's dark nature".  Whether or not Drac learns that Adrian is his half brother or not is up for grabs.  It could make for plenty of intrigue between the two later on, but it's also not necessary. Doesn't matter. 
Of course, maybe this route removes Alucard's powers as a dhampir (which isn't a big deal IMO)...or perhaps his blood relation to Drac causes him to take to the vampirism in a different way.  Either way is perfectly plausible.