To add onto this, I know the Shogun effectively ruled but my understanding is the Emperor would give his blessing or appoint the Shogun. If the Emperor was a figurehead, why did this matter?
The only long period of time I can think of is, I suppose, the Sengoku period which isn't all that long (and where the shogun faced similar problems). At that time, the various so-called Sengoku daimyō did see a meaning in approaching the imperial court and seeking appointments to court titles which had lost their practical functions a long time ago. Some also saw a meaning in controlling Kyoto, then the seat of both the imperial court and the Ashikaga shogunate. (This period is however not my field, but I'm sure u/ParallelPain has got your back!).
Of course, technically Japan was also not unified in the early centuries of the inception of the imperial state (court), but that's because it was not yet unified, and is therefore a different issue. Conquest takes time!
Concerning your shogun-emperor question:
Precisely because a "figurehead" is not conceived as "having no function whatsoever" but more accurately as "having the function of providing grounds of legitimacy to a political system." The figurehead doesn't "do" anything except legitimize the actual political practices of others: the arrangement is that of a full delegation of practical, administrative functions to others, and the institutionalization of such an arrangement reduces the act of delegation to a formality or something that happens "automatically." The shogun had legitimate power because it was either explicitly or implicitly understood that he, in some respects, derived this power from the emperor. For this reason, in 1867 the last Tokugawa shogun Yoshinobu formally returned his authority to govern to the imperial court; and interestingly, he abdicated the office of shogun ten days later, implying that the two things were, perhaps, not exactly the same thing.
Either way, You may want to look at past answers on the shogun-emperor relationship or about the emperor throughout history, such as:
https://www.reddit.com/r/AskHistorians/comments/je4pfs/how_did_retiring_help_enperors_to_be_more/
https://www.reddit.com/r/AskHistorians/comments/jwiz3l/how_come_no_shogun_ever_tried_to_usurp_the/
Likely, u/ParallelPain knows a couple more spots to direct you to.