The Dugbogs were not crushed by a strong Re’em, but by a Re’em’s strength. The strength was used to crush them, and the strength was what you got out of it.
The knuts were not forged by a hot forge, but by the forge’s heat. The heat was used to forge them, and the heat was what you got out out it.
In your scenario, the immortality of the phoenix was not used to burn the corpse, so you cannot get immortality out of it.
For strength, you use Dugbogs that were crushed by a strong Re’em.
For heat, you use bronze that was forged in a hot forge.
For immortality, you use a corpse that was burned by an immortal phoenix.
Speaking in terms of significance:
The Dugbogs were not crushed by a strong Re’em, but by a Re’em’s strength. The strength was used to crush them, and the strength was what you got out of it.
The knuts were not forged by a hot forge, but by the forge’s heat. The heat was used to forge them, and the heat was what you got out out it.
In your scenario, the immortality of the phoenix was not used to burn the corpse, so you cannot get immortality out of it.
K. I was confused because Bellatrix hadn’t died, mostly. Your edit helped.