I have found an official ruling in an analogous situation that agrees with my interpretation:
[...]
So from that answer, one can deduce that the Pillar of Flame can be applied first, replacing "the creature dying" with "the creature being exiled". Then, regeneration can be applied second, replacing "the creature dying" with "the creature becoming tapped, having its damage removed, and removing it from combat if applicable." Since the creature is still being destroyed as a result of lethal damage, regeneration's replacement effect still applies and the creature becomes tapped, has its damage removed, and is removed from combat.
But if one applies regeneration first, one replaces "the creature dying" with "the creature becoming tapped, having its damage removed, and removing it from combat." The creature will no longer have been dealt lethal damage, so the Pillar's replacement effect doesn't apply.
You may sacrifice any creature to activate the ability, including Bloodflow Connoisseur itself. Upon resolution, the ability won't be able to put a counter on the object, so it won't do anything.
So from that answer, one can deduce that the Pillar of Flame can be applied first, replacing "the creature dying" with "the creature being exiled". Then, regeneration can be applied second, replacing "the creature dying" with "the creature becoming tapped, having its damage removed, and removing it from combat if applicable." Since the creature is still being destroyed as a result of lethal damage, regeneration's replacement effect still applies and the creature becomes tapped, has its damage removed, and is removed from combat.
But if one applies regeneration first, one replaces "the creature dying" with "the creature becoming tapped, having its damage removed, and removing it from combat." The creature will no longer have been dealt lethal damage, so the Pillar's replacement effect doesn't apply.
Did I read that response correctly?