DoDI 1304.26[0] describes department-specific requirements, and points back to USC 10 for the legal requirements. USC10 S504(b) lays out the specific requirements, namely "... lawfully admitted for permanent residence[2] ..." and in-addition, as you point out, there exists program-exceptions performed or maintained by the SecDef. A report by the Deparment of Homeland Security suggests that some immigrant visas meet this criteria[3]. There's also a pretty long list of spouse-visas that constitute LPR[4].
So, it's still possible. A recruiter would know more, or at least know someone who knows more. Where there's a will, there's a way. Notice these are all primary sources, with a favor towards the authoritative information. If you're a stickler though, you could pull the PDFs direct from congress, but that's as much fun as reversing a box of punch cards.
[0] DoDI 1304.26 Encl. 3, 2(b) https://www.esd.whs.mil/Portals/54/Documents/DD/issuances/do...
[1] USC10 504(b) https://www.law.cornell.edu/uscode/text/10/504
[2] https://www.law.cornell.edu/definitions/uscode.php?width=840...
[3] https://www.dhs.gov/sites/default/files/2023-02/2022_0405_pl...
[4] https://www.dhs.gov/immigration-statistics/lawful-permanent-...