These two are distinguishable in speech, because when you mean #1, you are going to say the whole phrase in a single intonation block, but #2 will be two: 抑えられない／幸せにしたい気持ち, reflecting the structure that two chunks being parallel modifiers of the last noun.
If written, it is ambiguous in theory. I said "in theory" because most people would parse it in #2 if you showed them the line.
幸せ works both as a na-adjective and a noun. Na-adjective is basically a noun in form except limited particle connection, using な to modify nouns, and being adjective in meaning. If you parse like #1 i.e. 抑えられない modifies 幸せ, 幸せ needs to be a noun because adjective can only modify a noun with the dictionary form. As a result, it means:
(1') a feeling that [I] want to turn [something else] into an irresistible happiness
If #2, 幸せにしたい has no modifier and would be either na-adjective or noun, but na-adjective by default. Then it means:
(2') an irresistible feeling that [I] want to make [somebody else] happy
Which is a likelier situation? :)