The answer is we will never know, in the same way that we will never know for certain which bits of Hamlet are by Shakespeare and which bits are actors' interpolations, or which bits of the Henry VI trilogy are by Shakespeare and which by another author.

In any case, we are too picky when we need to be so particular about the very words of an author. Most modern prose works are heavily edited by the editorial staff at publishing houses, are they the 'real words' of the author? (Even poetry is increasingly being rewritten for poets by poetry editors). In the C18 MSs were very lightly punctuated and authors expected the printers to sort out their punctuation for them. Is the syntax of these works the author's or the publishers?

'A Lover's Complaint' appeared as a pendent to the Sonnets of Shakespeare in the 1609 publication. It is not an obviously incompetent poem, and seems in places to be a deliberate parody of bad love poetry, and more importantly it provides a foil to the Poet/Young Man/Dark Lady triangle of the Sonnets, with a much more common faithful woman/faithless man duo, almost as though the Sonnets are being normalised with a conventional pendent.

'A Lover's Complaint' was first published with the Sonnets, and has been discussed in conjunction with them by a small, but interesting strand of Shakespearian criticism in the C20. I think it would be remiss to leave the text out of a Complete Works of Shakespeare.