I doubt something as mundane as the killing of Alucard's heart would be the death of him. The one time he was nearly killed, it was because Anderson was wielding the power of Jesus Christ himself.
This was stated and implied by Alucard himself. The death of a vampire's heart is a stereotypical weakness... and Alucard has quite a few of those traditional weaknesses left in him. Killing his heart is one of them. I can prove this because Alucard (without any lives inside of him) was decapicitated by a very holy bayonet wielded by Anderson augmented with the Nail of Helena (which was blessed by JC) and regenerated easily. And yet a base Alucard (but with millions of lives) had to take a very different way from regenerating when a weaker Anderson stabbed bayonets into his heart (which would kill his heart) by constructing a new body from bats that appeared out nof nowhere. This was the only time shown when Alucard had to take this way out and the first time from the start of the series that Alucard lost a life/soul.