I think you can probably do it. I'd use Sulky cotton 12 weight since it comes in a spool and not in separate strands. I'd try to sink my beginning and ending knots through the batting if you can manage that. I'd make a little sample square prepared the same way as your quilt to practice on before you try the 'real' quilt'. If you finish it with binding, you can wash it the way you would the quilt and see how the enpmbroidery stands up to washing. Then you can switch to an embroidered label if you don't like the results.
Please let us know how this works out for you! And p.m. me if you have any questions.
I have often been a member of the better late than never club so you are in good company, as far as I am concerned.