The only grammatical fault with I can see with what you wrote is that you pluralised "birthdays". Aside from being uncommon to do so, if you were actually wishing that all future birthdays are happy it becomes a redundancy to then say "any many more". The other unusual thing about your statement is that you are wishing future ones be "happier", and that almost suggests that this current one isn't as happy as it could be. What I think you mean to say is that they will all be equally "happy".
"Happy Birthday to You... And Many More"
Many people conclude the "Happy Birthday" song by singing "any many more".
After the song is sung, party guests sometimes add wishes like "and many more!" expressing the hope that the birthday person will enjoy a long life.
If I were to correct what you wrote, I would say:
Happy birthday my dear... and may there be many more.