To be fair, Howletts/PL is not the only place perpetuatig this myth- these Lions, or the uncastrated ones anyway, are afaik still part of a managed breeding programme involving several other zoos in Europe too. And there are other zoos which have participated in the past and have since dropped out.
Good point about the Asian Lion- certainly a very close relative, if not identical to Barbary (as it carries a different genetic Haplotype) then perhaps as close as the Siberian and Caspian tigers, which apparently represented populations of the same species which have experienced seperation in quite recent times.
Yes genetic work shows north African was repopulated by Asian lions shortly after the original population mysteriously went extinct thousands of years ago. The original range of the Asian lion was north west India, Persia and the middle east/turkey.
mDNA evidence also suggests the Caspian tiger was also not a subspecie, but the eastern population of Siberian tigers, which originally had an extensive range from the amur region, north over mongolia, across the CIS states as far east as the Caspian sea. The separation of the east and west populations appears to have occured very recently in history.