From a halachic perspective, would the British refusal to allow Jews into Israel be enough to declare war and kill? No one was in mortal danger.

Historically, the British refusal to allow Jews to enter the Land of Israel, did (!) put them into mortal danger, by forcing them back to Europe, and many were murdered in the Holocaust. Also the anti-Semitic British prevented the Jews from protecting themselves from the Arabs attacks, which was an additional mortal danger. The halacha is clear: it is obligatory to fight those who cause Jews to die, based on Milchemet Mitzva, the law of Rodef, and "Thou shall not stand by your brother's blood" (Shulchan Aruch, Ch. M. 426; Or. Ch. 330; Sefer HaChinuch 600). In addition, throughout the Tanach and Jewish history, Jews went to war, inevitably killing and also giving their lives, for the sake of liberating the Land of Israel, that she should be under Jewish rule (including the Maccabees and R. Akiva's Bar Kochva rebellion which were even after there were no longer Urim v'Tumim, see Rambam Hil. M'lachim u'Milchamotehem 11, 3). Accordingly, Jews were not only allowed, but obligated to declare war against the British and the Arabs.