Once again, I had issues very similar to this when Pebbles was a puppy. She was so out of control when I brought her home (as a rehome) at 5 months that I could barely keep her in the sink. Her first grooming appointment she tried to bit the groomer and I was told to never bring her back.
But, I went home and decided that I was going to fix her grooming issued and I have. When I bathed her I did not muzzle but I did keep her harness on with a very firm grip. She was bathed weekly and brushed daily. I'm not saying that my methods will work with this puppy but I sure wouldn't let a puppy bite and draw blood over bathtime.
Maybe if I am more specific in my method it might help.
This is how I would prepare for a bath with Pebbles when she was small. It really helped and she is now very well behaved at the groomer as well as for her weekly baths.
I prefer to use the kitchen sink for baths. Since it's higher and (mine) has a sprayer I can get the job done quickly.
Gather all your supplies and place them within reach.
Make sure you have brush, comb, bands and bows set up in a separate area for grooming afterward. I like to use the top of my wash machine. I place a towel down and lay out all my supplies there as well. Make sure you have your blowdryer plugged in and read to use as well.
Place a small towel in the bottom of the sink so that puppy will not slip and feel secure.
With the harness on place the puppy into the sink with the water already running.
Hold onto the harness behind the shoulders at all time. She will not be able to bite you and you will have control over her movement. DO NOT let go!
Use the sprayer and hold it close to her body and start at the back of her head wetting her entire body all the way to the tail and under her body as well. Work with a purpose through the entire bath. DO NOT let go!
Rinse from top to tail and then apply conditioner and rinse.
Pebbles learned very quickly and after just a few times she did very well.
As for grooming afterward you can do the same thing. Do not try to brush or comb while she is on your lap. You need to set an area where she knows you mean business~grooming business! Always work quickly and with a purpose.
If you have questions you can always PM me and I will try to walk you through some of the other things that worked really well with Pebbles.
There is no reason for her to be able to bite you if you bathe her this way. Please don't continue with dominance it just will not work. You need to establish that everything she does needs to be earned.
Pebbles is not perfect and is still a guarder to this day and she is now 4 yrs old. But there are somethings that she is excellent at like sit/stay. There are certain commands that need to be establish NOW like darting out doors before you and gobbling her food. You should also work on these issues as well with her. I can open by front or back door and she will sit/stay (while quivering to bolt) until I tell her she can go. She knows to sit, paw, shake before she can eat too. All these things have help so much in getting many of her negative behaviors in check.
Please PM me if you have more questions. I want to help