ἡ πάλη is a noun, occurring only in Eph. 6:12 in the New Testament but a common Greek word. It has to do with a struggle or contest one is an active participant in. In contrast, swayed makes one a passive participant, and thus is less in line with the meaning of Eph. 6:12. A solder doesn't put on armor to be a passive participant in battle.
The wording οὐκ ἔστιν ⸀ἡμῖν translated literally is rather awkward in English (is not to us) but is not awkward in Hebrew (אין לנו) and is a Hebraism meaning we do not have, "we do not have the battle/struggle ..." The article means the struggle Paul has already started describing. Do not be would be expressed something like μὴ γίνεσθε (Eph. 5:17). Thus, we would expect "we are not to be" to have μὴ instead of οὐκ for the negative particle, and a form of γίνομαι instead of εἰμί for the verb.
Note: If you translate it "We are not to be swayed by flesh and blood," then it would follow that we are to be swayed by "the rulers, by the authorities, by the cosmic powers over this present darkness, by the spiritual forces of evil in the heavenly places." That would be a rather demonic interpretation. Paul is contrasting the physical with the spiritual demonic and a translation "we are not to be" loses that contrast.
Lexicons
πάλη, ης, ἡ (παλαίω ‘wrestle’; Hom. et al.; ins; Sb 678, 6) engagement in a challenging contest (orig. ‘wrestling’ Il. 23, 635 al., then of fights or battles Aeschyl., Ch. 866; Eur., Heracl. 159) struggle against, fig. ext. (Longus 3, 19, 2 of love; Philo, Sobr. 65 πρὸς πάθη π., here w. wrestling imagery), of Christians’ fight against powers of darkness Eph 6:12 (the opponent is introduced by πρός w. the acc. as in Philo above, but the context suggests military imagery).—DELG s.v. παλαίω. M-M. TW.
--
Arndt, W., Danker, F. W., Bauer, W., & Gingrich, F. W. (2000). In A Greek-English lexicon of the New Testament and other early Christian literature (3rd ed., p. 752). University of Chicago Press.
πάλη [ᾰ], Dor. πάλᾱ, ἡ, (πάλλω) wrestling, Lat. lucta, Hom., Pind., etc. 2. generally, battle, Aesch., Eur. --
Liddell, H. G. (1996). In A lexicon: Abridged from Liddell and Scott’s Greek-English lexicon (p. 587). Logos Research Systems, Inc.
P.S. Example of Hebrew grammatic structure
וְאִם־אֵ֨ין לָאִ֜ישׁ גֹּאֵ֗ל
(from Num. 5:8, BHS2003)
Literally "If there/it is no kinsman [redeemer] to the man..." But the meaning is:
If the man has no kinsman ...
(from Num. 5:8, JPS1985)
Greek:
Our wrestling is not (οὐκ ἐστιν ἡμιν ἡ παλη [ouk estin hēmin hē palē]). “To us the wrestling is not.” Παλη [Palē] is an old word from παλλω [pallō], to throw, to swing (from Homer to the papyri, though here only in N. T.), a contest between two till one hurls the other down and holds him down (κατεχω [katechō]). Note προς [pros] again (five times) in sense of “against,” face to face conflict to the finish.
--
Robertson, A. T. (1933). Word Pictures in the New Testament (Eph 6:12). Broadman Press.