I'm aware that Christianity had some influence before Japan was closed off to most foreigners, but how ddi it persist during the Edo period? If the Tokugawa shogunate suppressed Western influences, including Christianity, so much, why did many Japanese people still continue to embrace it despite persecution?
I think you grossly overestimate Christianity's presence in Japan in the Edo period. After the Shimabara rebellion in 1638, Christianity was ruthlessly stamped out by the Shogunate, with practitioners usually being exiled to the Philippines or other countries, aside from a tiny minority that secretly practiced Christianity, mostly notably in Nagasaki.