Hi, Follow these general guidelines. Capitalize the phrase when you simply say the two words to someone, eg 'Happy Birthday, Mary!'. Don't capitalize it if it is part of a sentence. eg 'Mary had a happy birthday yesterday.'
Given that 'birth' and 'day' are two separate and unrelated words; as are 'remembrance' and 'day', for example, and, as with 'Remembrance Day', 'Birthday' is a combination of these two words and thus a given name to a specific event: i.e. someone's birth, so should 'Birthday', as a name, not also be treated as such, recieving a capital letter to start, whenever used, even in everyday conversation
I appreciate the sentiment actually, having 'happy Birthday' does seem somewhat abnormal, though, looking at it here it's hard to tell. It looks fairly regular, looking at it now. But it is a tough issue!