The Americans with Disabilities Act covers schools and nurseries, so your son's kindergarten has to comply with it. I'm not a lawyer or doctor, but a gastroenterological issue looks like it would be covered, and the rules in the ADA say that they should be interpreted as broadly as possible.
Getting you to sign a piece of paper saying that you volunteered to withdraw your son sounds like a nasty way of tricking you out of your rights. DO NOT SIGN IT!
I suggest you start by reading the ADA. Print it out, highlight the relevant bits, and take it to the next meeting. I suggest starting with:
Definitions: look under "Place of public accommodation"; schools are listed there, so the ADA applies.
"Definition of Disability": digestive issues are covered. Lack of bowel control is a limitation of a "major bodily function", and would be considered a disability if it is the result of a medical issue. On the other hand an inability to learn to use the toilet at the normal age would fall under the related "mental impairment" rule (assuming that usual toilet training has been attempted). Inability to use the toilet like other children means that your son cannot care for himself to the extent that would normally be expected, and this is a substantial limit in a major life activity. "Learning" is also a major life activity that will be substantially limited if he cannot attend school.
"Modifications in policies, practices, or procedures." They have to do anything "reasonable" to accommodate your son, and if their existing policies get in the way then they have to change. If they haven't got the resources then they need to get some more (within reason), and they can't charge you extra because of it.
Also check out the section on "Retaliation". If they threaten to blacken your son's school record over this, that is also breaking the ADA.
The biggest headache here is the word "reasonable"; ultimately if both sides dug their heels in it would be for a court to decide. However your proposal of having a family member on-hand as an assistant certainly sounds more than reasonable, especially if there is a precedent (do ask "How is my case different from that?").
You say "some weeks as high as 2-3x". If you mean this is only happening 2 or 3 times a week then just telling the school to deal with it would seem reasonable to me. If you meant 2-3 times a day for a week, then maybe less so. A lawyer will be able to give you a better feel for how courts view cases like this.
One other thing: keep a log of communication with the school. If they invite you to a meeting then keep notes, and as soon as possible send them your summary of the meeting including anything they promised or threatened. Same for phone calls.