From a rescue point of view, I can say immediately that a good rescue will do everything possible to support you and to help you do the right thing by your dog. People in rescue know that life does not always follow a script, and that things can happen which make the most secure life turn upside down. As I write this, there are two dogs inches away from my feet that were initially surrendered for emergency foster care (and again a good rescue will help if you simply need temporary assistance). After a time, it became impossible for the owner to have them back and they are now much loved members of our family. Seeing the process close up taught me a lot - in particular surrendering owners will often have to think long and hard about what is best for the dog and sometimes surrender is the answer.
On a practical level, there are things that you can do to make the process work smoothly, especially after a biting incident. Give the rescue a full history. If you surrender, please also give your dog's paperwork such as vaccination certificates, chip details, any vet records, training etc. - this can save unnecessary duplication and expense for the rescue. It's also worth preparing a short "crib sheet" about your dog's daily routine - things like exercise, grooming, favourite toy, any allergies, level of training (be honest!) - this sort of info is invaluable if the dog is going to a foster home, and will help speed up their journey to a new home. Also include info on diet - how often, what sort of food etc. Again, this is tremendously helpful if your dog is going to foster care.
Please don't expect the rescue to swoop in and remove the dog within 24 hours of contact (a lot of people do). Most rescues are run by volunteers and usually for surrenders a web of transporters, foster homes and/or kennel spaces need to be found - this can take a little while. If you are willing to be flexible and accommodating about travelling to meet transporters/foster homes etc this is also very helpful.
Wishing you all the best.