> Immigration can't be effectively regulated by any means available to a moderately democratic state.
That's debatable. But lets assume it for the sake of argument.
> Should immigrants receive the BLS? If yes, you go broke. If no, the immigrants are essentially second-class citizens who have to work for an extremely low wage (because they would have to pay indirectly for the BLS).
I think this claim is overly simplistic; also, I think there are several categories to consider,that I think you've failed to distinguish:
(1) Naturalized citizens should certainly receive whatever universal or guaranteed income is adopted, but immigrants don't become naturalized citizens immediately.
(2) Legal immigrants (the number and characteristics of whom a democratic state can effectively regulate), who are not citizens, probably should be eligible for the support [1] as well, but may be admitted on criteria that include having means of support that make it less likely that immigrant will be a recipient (in a GMI scheme) or net recipient (in a BI scheme).
(3) Non-immigrant, but legally-present workers, visitors, etc. should not generally be eligible for the support. In the case of non-immigrant economic admission categories, adequate support means should be part of the conditions for admission. There may be some exceptions to this (note [1] again may apply to these exceptions) or some categories of non-immigrant admits that are qualified for alternative, status-specific support.
(4) Those illegally present probably should not be eligible for the benefits, but may be eligible for some kind of normalization of status process after which they would become legal immigrants who might be eligible for benefits (to which note [1] would apply again). This normalization process might include up-front payment of a fine for those with means, or commitment to income contingent future payment of the fine with nominal interest, and might even include limitations on benefit eligibility for some period even beyond what regular legal immigrants might face, or commitments to repay any benefits claimed during a certain transitional period on an income-contingent basis. (In general, payment for normalization -- or payment for above-quota legal immigration -- makes sense for individuals who are not personally barred, since limits on levels of such immigration are a cost-of-immigration control measure, and its better for people who want to immigrate to bear the costs then for society to bear the costs of inefficiently trying to keep them out, and then also bearing the costs of them coming in anyway.)
[1] As productivity advances and if the basic support improves to provide more than a very minimal floor, its not entirely unreasonable that non-citizens might recieve less than the full support level, however, possibly based on a sliding scale based on residency; in theoretical terms, in a rent on the commons model, this represents a kind of "vesting" process where the immigrant doesn't become a full "owner" until they are naturalized; in practical terms it provides basic social support but reduces the immediate immigration incentive.