***** NOT A TEACHER *****
Of course, I do NOT have the confidence to say whether it is correct or incorrect.
But I can tell you the following.
1. Google has some (not many) examples of "___ slept weeping."
a. "I fell into a deep depression. For a year, I did not care whether I lived or not. I slept weeping, and weeping did I wake."
b. "Weeping and creeping, I slept."
NOTES:
1. In 1a, there is no comma, for "weeping" is part of the predicate. If you believe the scholar George Oliver Curme, the participle modifies the verb "slept."
2. In 1b, the comma shows clearly that the participle is modifying the subject.
3. 1a comes from A Day in the Life of a Smiling Woman (2011) by Margaret Drabble.
4. 1b comes from In the Valley (2013) by Carmen Beck.
5. Students who want to understand Professor Curme's reasoning will find it on page 43 in Volume I of his 1931 masterpiece A Grammar of the English Language.