I don't know much about bobtail either.
But for any recessive gene, if you cross one parent who has it (your rooster) with one parent who does not (hen), you get F1 chicks that carry the gene. Breeding them to each other gives F2 chicks with about 1/4 showing the recessive gene. Or doing a backcross of F1 chicks back to the parent gives about 1/2 chicks that show the recessive gene.
When you are working with more than one recessive gene, the numbers get worse pretty fast. Two recessive genes would appear together in about 1/16 of chicks in the F2 generation, or 1/4 of the chicks in the backcross generation. Three recessive genes would be expected to appear together in only about 1/64 of F2 chicks, or 1/8 of backcrossed chicks.
When there are sexlinked genes involved, it can be easier or harder depending on which gender has which genes.
If you have the option of black cochin or blue cochin hen, I would use the black one (because then you don't have the blue gene making things any more complicated than they already are.) If you have to use a blue hen, I would cross her to the bobtail rooster and only breed from chicks that show black (not blue).
As regards information specifically about bobtail, I found this:
https://www.cochinsint.com/uploads/8/8/2/1/8821267/ci_-_bobtail_cochins.pdf
It's from 2013, and I can't find any updates on the breeding trials that are mentioned.
If that is right, the actual bobtail gene should be a sex-linked gene, and I think the "blackred" color they mention might be Birchen chickens with red instead of the silver.
A Birchen hen might be a good choice, although it could be even better if you found one showing gold or red instead of the silver (I'm not sure of the correct variety name in Cochins: maybe Gold Birchen or Brown Red?)
If you find the "right" hen to work with, and if the actual bobtail gene is sex-linked, you would get bobtail females in the first cross to your rooster, and breeding them to their father would give pure bobtails. Breeding the F1 pullets to their brothers would give some bobtails in both genders, perhaps as high as 50% bobtails.
Note, the previous three paragraphs involve some guesses and assumptions, and I may be wrong in one way or another. If I am wrong, hopefully someone else will set us both straight.
@nicalandia do you know more about bobtail? Both the gene, and what other genes are required (E^R Birchen? does gold vs. silver matter?)