Is it possible to nest VPNs, i.e. connect to a VPN in the location of the preferred fake home, then connect to Facebook's VPN through that? I genuinely know enough about VPNs to know if this is possible or not.

Yes. This kind of thing is generally called "tunneling", not "nesting". As applied here, "chaining" would make more sense.

Oops, I meant to say "genuinely don't know enough about VPNs". Too late to edit now...

That's only an issue if the use of a non-jailbroken/rooted phone is a requirement for doing the job. "I like my phone rooted/jailbroken" is a plausible excuse already.

They provide a biz phone for on-call and jail-breaking it is probably out of scope of what is permitted.

If you try hard enough, there are mobile carriers that will hand over the SIM's data session to a tunnel endpoint of your choice (via L2TP or IPsec).

There is always a way.