Can a non-resident alien with an ITIN (for work purposes only) use this to open a single member LLC or do they need to apply for an FEIN with the IRS?
Thank you.

They need an EIN since they will be taxed as a corporation.

§ 1.6038A–1 General requirements and
definitions.
* * * * *
(c) * * *
(1) * * * A domestic business entity
that is wholly owned by one foreign
person and that is otherwise classified
under § 301.7701–3(b)(1)(ii) of this
chapter as disregarded as an entity
separate from its owner is treated as an
entity separate from its owner and
classified as a domestic corporation for
purposes of section 6038A. See
§ 301.7701–2(c)(2)(vi) of this chapter.

Thanks - what is your opinion on an SSN card - saying For Work Purposes Only - owner applying for a mortgage? To me, this is too high of a risk in dealing with a borrower that has a finite time to stay in the US. US employment Authorization card expires in 2 years or more - says not valid for re-entry to the US. Getting some pushback from lender.

That is a credit decision not a compliance one. Where is the future income coming from?