Translating あくまで一応ですけどね Translators say: means: doggedlyinsistentlypersistentlystubbornlyto the bitter endto the endto the lastto the utmost means: tentativelyfor the time beingin outlinejust in caseonceone timeso far as it goes I couldn’t understand how to solve this puzzle

It's very hard to give a correct translation without any context, but usually means something like "You know, this is just for making sure", "Well, I know this is not mandatory/critical", "I'm suggesting this merely as a backup, after all", "I don't have a strong opinion, though", etc., depending on what is actually referring to.

is a difficult expression, in this context it's closer to "just" and/or "after all". Basically it's emphasizing the meaning of . See: Meaning of in