Such as arctophile or entomophile? Looking for something more formal sounding than 'cuteophile' or something similarly made up.
While not a perfect fit, a person who likes beautiful things would be an aesthete.
In Japan — the home of all things cute and cuddly-wuddly — the term kawaii, meaning cute, lovable, adorable, etc., "has become a prominent aspect of Japanese popular culture, entertainment, clothing, food, toys, personal appearance, behavior, and mannerisms". It has now entered the English dictionary as a loanword. People often talk about the "kawaii syndrome" affecting all of Japan. That said, a kawaiiphile sounds just as made up as cuteophile. Depending on context, you might be able to stretch kawaii-otaku (otaku is another loanword) to imply an obsessive interest in cute things.
Then there's agalmatophilia which involves "sexual attraction to a statue, doll, mannequin or other similar figurative object".
According to Perseus the Ancient Greek φιλόλιχνος, philolichnos meant "loving dainties, lickerish, dainty", so turning it around we'd get: lichnophile.
In Modern Greek, according to a correspondent at wiki.answers.com, "Cute is χαριτωμένο (haritomeno) which can be applied to both soulless objects (teddy bears, dresses etc.) and living beings (humans, dogs, cats etc.)", which would yield charitomenophile.
It would be interesting to know whether lichnophilia (or charitomenophilia) is clinically related to Acquired Cuteness Disorder.
There are lots of names, but if you're looking for a name like "otaku" or "weeaboo", there are sadly no names. I love Kawaii things myself, so I would suggest making up your own name. Try "Kawaiiboo" or "Kawaiiku".