Call me cynical, but I believe that those concerns are totally unfounded because the "evil government" doesn't need your FB login for that, IMHO they most likely can access that information directly without your consent. I would be really surprised if they really had to do this mass harvesting of profiles based on a series of one-time access to individual profiles of a small fraction of population; They had much more effective ways to do mass harvesting many years ago (as illustrated by various leaks) and probably do them now, having to involve thousands of immigration agents would be simply counterproductive.

I do believe that immigration services look at your account exclusively for their goals of evaluating immigration. They ask you to unlock your account because for them it's objectively simpler to ask that information from you than to get it from the other government agencies that have that information but won't disclose it because formally they shouldn't have it. There's no "the government", it is an amalgamation of many organizations that have conflicting goals and don't coordinate, so it's reasonable to expect that "the government" has some information and at the same time "the government" (i.e. other parts of it) don't have it and need to request it from you or do without.