You may have seen that Ancestry is offering free access to their site this weekend. They have records of people emigrating from England. Find My Past likewise but you will have to pay to see their records unless you are prepared to wait until they offer a "free" weekend. I do not know where you live but, in UK, free access to both sites is often available at public libraries, although some have now withdrawn that facility as an economy. I believe similar arrangements exist in some other countries. It costs nothing to ask.
Another useful site is the Statue of Liberty -Ellis Island Foundation which has comprehensive on-line records of immigrants to America.
Not knowing the name JLB was using will make things difficult but it is possible to focus your search.
Hopefully he adopted a name not too dissimilar from his previous one. So he could have continued to use John and/or Lewis-Louis and some modification of Blackbeer -Black, Blackman, Beer, Beerman, etc. Possibly Weinberg. Do you know his original Lithuanian name? He may have used that or an English translation of it or a similarly sounding English name.
I suspect he left shortly after the birth of his youngest child. You said he had four although I can only find records of three. If Sadie was the youngest I would focus on the period 1919-21.
You know he was a tailor, born in Lithuania in 1892 (does his age on his marriage certificate match?) It is unlikely that he would have fabricated any of that. You mention that at some time he was in Brooklyn which suggests he had contact with his English family after he left for America. Do any of your older relatives have any other half-remembered stories about him? They could provide other important pieces of information.
If you find any names on passenger lists that seem to be a close match, see if there is any more information about them on Family Search to help fill out the individual's personal profile. This is where family stories could provide some corroboration.
Good luck. You have a serious challenge but you must believe that there is a chance you could strike lucky.