I'm not sure support will even noticeably dip when Charles is king. Before that happens there will be a huge sequence of events kicked off by the Queen's death, which will include a big funeral, a mourning period, wall-to-wall-to-wall-to-wall-to-ceiling-to-floor media coverage that redefines the concept of media "saturation", then eventually a coronation (months later - it's not out of the question that it might be more than a year afterwards). That'll come with its own coverage and an approximately infinite amount of themed paraphernalia, plus there will be the usual state trappings as the UK's anthem changes to God Save The King, Charles' face starts appearing on newly minted currency, and all the UK's ERs become CRs or GRs or whatever. The media narrative will probably be along the lines of Charles, the man born to be king, finally fulfilling his destiny.
There's every chance that that prolonged celebratory and commiseratory process will boost the popularity of the royal family in general and Charles in particular.
There might not even be a Charles III. Charles has grown up fully immersed in this kind of thing. He is certainly aware of the associations and also aware that he can choose a different regnal name to avoid those associations.