In the 1937 edition of The Hobbit, this word phrase does not appear.

In the OED, it has a compound word entry “birthday present” with an example of “birth-day present”, but in no case hyphenated as JRRT uses it here.

• 05.085 My birthday-present!’
• 05.087 My birthday-present!
• 05.109 My birthday-present.’
• 05.111 My birthday-present!

“birthday, n.” OED Online. Oxford University Press, June 2017. Web. 5 September 2017.