Welcome to he toilet teaching board. I'm sorry to hear that you are having so many issues with your son.
We have seen the Dr., what his explanation is same as Encopresis, though he never really said Encopresis..
You might ask your doctor to order an x-ray of your ds (dear son) abdomen to see how much poop is in his intestinal tract. If missing only one or two doses of Miralax causes him to have an accident, then he is probably too full of poop and needs to be on a higher dose of Miralax.
Has your ds always had problems with pooping, or is this a new problem since you began potty training? If it's a problem only related to potty training, then he may be withholding his poop which is causing the encopresis. If he is withholding his poop, then your ds will probably need to be on a much higher dosage of laxatives to make it impossible for him to withhold his poop.
How does your ds react when he's had an accident? Does it bother him? Are you having him help clean up the accidents?
You may need to take a spare change of clothes for your ds to keep in the nurse's office or kindergarten room at school in case he has an accident. You can also buy wristwatches with a vibrating alarm to discretely remind ds to go to the restroom while at grandpa's or school (bedwettingstore.com has some vibrating watches, tho' you can probably find lower quality versions elsewhere for less money).
Although it may be hard, your ds definitely needs to cut back on the amount of milk he drinks. Besides being a potentially constipating food, nutritionally he shouldn't be getting such a large quantity of milk, either. You may need to dilute the milk to make it easier for him to wean off from it.
I have asked a couple times if we shoudl do comething to check things out, but so far no Dr has done any tests.
I'd push the doctors harder. My 14yo dd has had bowel troubles since birth, but it wasn't until she was 4yo that the doc reluctantly suggested laxatives and until she was 9yo that a pediatric GI (gastroenterologist) doc finally diagnosed encopresis. It was a couple of years after that before a urologist ordered an x-ray. At that point, the GI doc could finally see how bad dd problem was and began to treat it more aggressively.
I have learned over time, from both my own experience, and from listening to countless other people share their experiences, that many regular pediatricians aren't familiar enough with encopresis to treat it properly. And, even GI docs don't always make the right call. Time after time the GI felt dd belly and told me she wasn't retaining poop when it turns out she was backed up to her eyeballs full of poop.
Even this past week the GI felt dd belly and said it didn't feel like dd was retaining poop. I asked the doc to order an x-ray to be sure, and the doc was surprised to see how much poop really WAS in dd intestinal tract. Before the x-ray, the doc thought we were using too much laxative already (dd was reporting that her stool was very soft and that she was pooping several times a day). After the x-ray, the doc ordered increased use of laxatives since it's clear dd is retaining too much poop.
An x-ray is the ONLY way to be sure whether/how much poop is in your ds intestinal tract. Be as insistent as you can about getting that x-ray, so the docs can see how significant your ds problem is. It's very likely that your ds needs to be on a higher dose of laxatives.
In addition to getting an x-ray, get your doc to write a note identifying your ds encopresis as a medical condition (which it is -- this is not something your ds currently has control over). The schools have to provide accommodations for disabilities. My dd has bathroom accommodations at school.
Also, read my response to the post titled "OMG somebody help us." I didn't see your comments there until I'd responded to this posting of yours. Some of what I posted in the other thread may be useful info for you to better understand encopresis.
BTW, if your ds soiling is due to a long-term ongoing problem with encopresis and is not behaviorally based, then I wouldn't necessarily have him help with the clean-up (at least not as a punishment -- if you want to teach him how to clean himself after an accident, that doesn't hurt). Helping clean-up won't deter his pooping since the pooping is not within his control.