Meaning and use of phrase "proverbial bucket"

Whenever you see a "proverbial " it generally refers to some proverb or idiom. In this case, the idiom is a bit complicated by the proverbial use of the term "bucket" -- as a pun. Normally, one carries certain things in a bucket. To complicate matters, one other use of "carry" is being able to faithfully sing a song according to the proper tones. So one might "carry a tune", which means sing it with the proper notes. Of someone who is bad at singing you might say "he can't carry a tune." If he is really horrible, you might say "he couldn't carry a tune, even if he had a bucket to carry it in." This is a pun, and is sometimes expressed as "he couldn't carry a tune in a bucket", and it is a well-known way to humorously say someone is a horrible singer.

So we come to the proverbial bucket.

"The proverbial bucket" refers to the proverb (or idiom) about carrying a tune in the bucket, and then to say that such a bucket "has not been constructed that would carry my pitiful attempts" extends it still further -- that even if a bucket could carry a tune, in my case it would have to be a very very extraordinary bucket before the tune could be carried by me, and such a bucket has yet to be constructed, and perhaps could never be constructed.

In short, your proverbial phrase is meant to suggest that the speaker is singularly inept at singing such that there is no hope in this world whatsoever that he could actually sing a song accurately.

I hope this helps.

ETA: It is worth pointing out (with a nod to @FumbleFingers comment to the question) that there is more than one "proverbial bucket". The other bucket is the one you kick when you die. Since we hope that singing is not a life-threatening activity, the bucket in this case is one we might want to carry a tune in. There are only two "proverbial buckets" that I am aware of.

ETA: It looks like I stole this answer from @DanBron, but I only read his comments in full after I posted this. I did comment about his prolixity without his actually posting an answer.