I think people really overestimate the importance of that record. Ultimately, its the goals scored that are much more important, not the record itself. Holding the record is only one part of Alan Shearer's legacy. It's not like if Kane breaks the record people will just forget about him. And for Shearer himself, I have no doubt that he'd rather give up the record than give up his PL title.
Kane is also in a very difficult position as someone who hasn't won a single trophy. It is remarkably unusual for a player of his stature to literally win feck all over their entire career. That a player like Le Tissier is the closest analogy really says it all. If he finishes his career without any trophies, even if he smashes the PL record, the fact that he won nothing will be a central aspect of his legacy. Written on his PL tombstone will be something like "Harry Kane, the most prolific PL striker of all time and also a player who won nothing and never had the ambition to challenge himself and go to a club where he could win trophies." He is a competitive, top level athlete - that simply has to grate on him badly.
Whether he breaks the record or not, he'll always be one of the greatest PL strikers of all time. Erasing that second clause from the tombstone - and just experiencing the joy of winning something big - is surely much more important to him than whether or not he happens to pass Shearer.