To my knowledge no serious theologian who doesn't believe in the immortality of the soul uses examples such as Lev 21:11 as proof-texts, so those who do (believe in the immortality of the soul) have not frequently felt the need to defend it on such a charge. Nevertheless, mainstream scholarship provides ample explanation of the lexical phenomenon.
The claim is made in the question:
the Hebrew literally reads "dead soul."
Although it's true that "soul" is the traditional rendering of nephesh, the term rarely (if ever) means "soul" in the specialized sense that it is used in Christian theology.1
The more vaguely and naively ["soul"] is used, the more correct and appropriate this translation becomes.2
Lexicons all agree that "soul" is applicable in a minority of cases; the only real disagreement is whether it is ever applicable, and nobody counts Lev 21:11 among the possibilities. Instead, along with a handful of other passages (Lev. 19:28; 22:4; Num. 5:2; 6:11; 9:6, 7, 10, 11, 13; and Hag. 2:13), this usage is glossed by HALOT (the standard modern lexicon of Biblical Hebrew) as:
deceased person, corpse.
We could leave it at that, but for the sake of thoroughness I will point out that the venerable TDOT rejects this definition (Horst Seebass). To understand why, we return to the OP's suggested translation "dead soul[s],"3 henceforth "dead naphshot" (= plural nephes). In fact, a more fastidiously literal translation would be "naphshot of [one] being dead" or "naphshot of a corpse". At this point we can provide an alternative response to the OP's query:
nepeš cannot refer to the corpse itself, but only to something associated with it.2
Although he "spiritualizes" nephesh in some sense, then ("vital force"), Seebass points out that all of these examples, which describe the uncleanness that results from proximity to the "naphshot of the dead", still do not carry the sense suggested in the question:
"Rich and abundant though this use of nepeš for life is, we must not fail to observe that the nepeš is never given the meaning of an indestructible core of being, in contradistinction to the physical life, and even capable of living when cut off from that life." We may therefore assume that even in death the vital force is something uncanny, belonging solely to God, which God’s people should not come near.4
1. The translation "soul" probably came about by way of the Greek psyche. See also, this Q&A which revolved around the same misunderstanding.
2. Horst Seebass. נפש, Theological Dictionary of the Old Testment Ed. Johannes Botterweck, Helmer Ringgren, Heinz-Joef Fabry, Trans. John T. Wills (Eerdmans, 1974) Volume 9, p. 512ff.
3. I presume this was intended, as the word nephesh is plural in Hebrew and in translation.
4. Seebass, p. 516, quoting Hans Walter Wolff.