Struggling with sentence of poem
Solution 1:
You are right with your suggestion. Come the morning is just a phrase (not so much in common usage today) meaning 'when the morning comes'. The join 'come the' is followed by a noun then used as an adverbial phrase e.g. come the morning, x, y and z will happen.
So in your context, it means 'when the morning comes, you flee my dream'.
This is in regards to the second definition of 'come':
occur; happen; take place
From Oxford Dictionaries
You could think of it as 'happen the morning...'