Here's how Gesenius explaned it:
(c) In the genitive governed by a substantive (cf. § 130 d), Ex 4:13 שְׁלַח־נָא בְּיַד־תִּשְׁלָֽח send, I pray thee, by the hand of him whom thou wilt send, i.e. by the hand of some one else;
Gesenius, F. W. (1910). Gesenius’ Hebrew grammar. (E. Kautzsch & S. A. E. Cowley, Eds.) (2d English ed., p. 488). Oxford: Clarendon Press.
That is, יַד is construct in בְּיַד־תִּשְׁלָֽח. Thus, a literal translation is "Please send by the hand you will send." For the rest see above ("i.e. by the hand of some one else."). The note in the JPS Tanakh says the literal translation is "send through whomever you will send" using the grammatical relative clause.
Davidson explains "by the hand of" meaning "through."
- Ex. 4:13 שְׁלַח בְּיַד תִּשְׁלַח send through (by the hand of) him-whom thou wilt send.
Davidson, A. B. (1902). Introductory Hebrew grammar Hebrew syntax (3d ed., p. 192). Edinburgh: T&T Clark.
§ 25. A clause may occasionally take the place of the gen. Such a clause will be what we call relative (in Heb. rel. or descriptive), particularly in designations of time and place: Gen. 40:3 מְקוֹם אֲשֶׁר יוֹסֵף אָסוּר שָׁם the place where J. was confined; Is. 29:1 קִרְיַת חָנָה דָוִד thou city where D. dwelt; 1 S. 25:15 כָּל־יְמֵי הִתְהַלַּכְנוּ אִתָּם all the days we were conversant with them; Ex. 4:13 שְׁלַח־נָא בְּיַד־תִּשְׁלַח send by the hand of him whom thou wilt send (send by means of some one else); 6:28 בְּיוֹם דִּבֶּר יהוה on the day Je. spoke.
Davidson, A. B. (1902). Introductory Hebrew grammar Hebrew syntax (3d ed., pp. 34–35). Edinburgh: T&T Clark.
. 10. person ⇔ hand — the whole human person conceived of in terms of the hand; perhaps with a focus on agency. Related Topic: Hand.
Ex 4:13 שְֽׁלַֽח־נָ֖א בְּיַד־תִּשְׁלָֽח׃
Brannan, R. (Ed.). (2020). Lexham Research Lexicon of the Hebrew Bible. Bellingham, WA: Lexham Press.