Rabbi Kaplan translated Derech Hashem, and seems to have explained things here as Ramchal explains them in Part 3:1:1-2, i.e. that the Vital Nefesh of a human is of the same kind as animals, but different in many respects, such as imagination, memory, intellect and will (similar to Rabbi Kaplan in his longer quote). The Vital Nefesh is the feeling and intellect of the creature, the life of the creature, unique to its species.
Ramchal also enlightens us there (3-5) to the fact that you and me are unaware of the presence and reality of the other soul we have that you allude to in the question; the higher spiritual one. This is because we are totally locked to the point of view of our Vital Nefesh, and that's the specific "who we are" he is talking about - the one that we experience in this life. The Vital Nefesh is who the Higher Soul is encouraging (through occasional unnoticable prompts of thoughts, images and will) to do Mitzvot, and who is earning Olam Haba in so doing.
That's who the Ramchal explains will be resurrected and live forever, but transparent to the higher soul that will shine brightly in Olam Haba (ibid 1:3:9-10).
On this topic of the two souls, Ramchal's explanation isn't hard to fit with other attempts to explain it that I've seen; it's not particularly controversial. Therefore B'EH my attempt at explaining Rabbi Kaplan - something impossible to do with certainty - is a good educated guess:
So yes, the nefesh Rabbi Kaplan is talking about can be immortal.
According to what I've seen in chassidus, the nefesh is indeed naturally mortal, but earns immortality and Olam Haba by Torah and Mitzvot, and connecting to Hashem through our Higher Soul, who seeks to do them and connect to Him by their very Godly nature. It also explains that this "lowly" world is going to become higher than the heavens at that time, and that this "lowly" nefesh is very important to Hashem, to say the least.