(there is some controversy over oatmeal and wheat/ gluten, so I just avoid it)
Yes, until recently oats were considered just as much a food to avoid by celiacs - the four grains that contain gluten are wheat, rye, barley and oats.
However, in the past few years it has been discovered that the gluten in oats are almost completely from contamination by other grains grown in the same fields or processed in the same factories. Oats grown completely in their own fields and processed in their own factories contain very little gluten. The UK (or Cdn? can't remember which) provides limits on how many completely "clean" oats a celiac should eat per week, and the limit really is quite low. However, some celiacs are more sensitive than others and can't even handle trace amounts of oats. I would say to avoid them completely, but it is a good gluten-containing grain to start with when you decide to start out introducing gluten. I would then go to barley, as it is quite low on the allergen scale, and then to rye, and wheat last as wheat itself can be allergenic (as in, the wheat itself is the problem, not the gluten in the wheat).
My LO's diet is quite similar to yours, as he is MSPI, and I kept him gluten free until 10 months. He always has had a very fragile digestive system and I've been worried about celiac but am trying my hardest to "find" the problem is elsewhere in hopes that it is not celiac at all. He is the same in that he will go a few days constipated and then have about a week of horrible explosive massive BMs. I can think of many occassions when my DH has changed a nappy and said to me "Honestly, I'm telling you, it was a LOG. I'd have been proud of myself for that one...."

Men. Sigh.
So a a couple of other things I've thought of looking at your LO's diet, only because they are observations that I have noticed in my own LO.
eggs -- we still only do yolks here and seem to be ok
avocados -- am pretty sure they give DS softer poops, I do try to give him a lot to get fats up
brown rice -- when he would eat it, very constipating
rice pasta -- ditto, the few times i tried, 2 pieces of rice pasta resulted in 2 days of no poop and then 3 days of hard lumps
beans (of all sorts. . . ) -- only done green beans and we are fine on them
pears -- DS eats one a day, days or weeks he misses pears don't seem to affect poop
peaches -- ditto
blueberries -- a small handful (8 - 10) has the same result as rice pasta, we avoid them like the plague unless he has diarrhea
grapes -- fine
prunes -- only used when consipated and they soften stool but do not have the "instant" effect that most moms say they will
bananas (very little; less than a half a very ripe banana a week) -- i use bananas as needed to manage poop! very ripe when he is constipated and less ripe when he's got diarrhea
green beans - see above on beas
rice flour bread - never tried it, but do rice flour pancakes and they are fine
gluten/dairy free waffles - never tried
gluten/dairy free cookies - never tried
sweet potatoes - used to eat them as his main carb (and squash) and aside from orange poop, no problems
potatoes - he won't touch them!!
olive oil - i add to food or fry with when i can, and he's fine with it
mandarin oranges -- these are fine for DS if I get them tinned, he can eat about 1/4 of a can every second day and be okay. any more than that and we get diarrhea. i have switched to fresh clemintines/satsumas now that they are available and he has had the most disgusting horrible awful diarrhea in the world, and i am wondering if they are extremely laxative
applesauce (limited) - i used to do this a couple of times a week and he was fine as long as there were no other binding foods like bananas in the same day
peas - fine, but they come out whole
lentils - never tried, i've never eaten them myself so not a clue what to do with them!
carrots - we don't really have probs with carrots, DS doesn't like them sooooo...
corn (took 4 days to go through his system: seems long; same with he lentils) - i find the same with corn, it takes 3 or 4 days to get through his system and when it does, poop is soft, icky and the corn is whole. and there is a LOT of it. yuck
Not sure if that helps at all or if it's pointless. I'm just thinking that our LOs seem to have similar digestive issues and VERY similar diets. My paed suggested that DS's back'n'forth bowel movements are because I limit his diet and that as soon as I get him onto table food without questioning what he's eating, that his poop will sort itself out. Paed really pushing me to get him onto soy now though, but wants me to wait until he's 2 for milk/dairy. The day I started allowing products that contained soy was also the same day that I started buying clementines fresh instead of tinned. Within 24 hours his poop went to unbearable disgusting diarrhea. I cut soy out about 4 days ago and have not noticed improvements, so today eliminated oranges as well. Will see how we get on. It will be easier for me to live without oranges than to continue soy free so my fingers are crossed!
The only things I can think of that my LO eats in addition to yours, that are still milk, soy, gluten free--
salmon
butternut squash
kiwi
mango
meat - DS loves ham (ie/ sandwich ham), and will eat most other meats incl chicken, beef and turkey, but i do find that they are quite constipating for him as well, so again, i try to balance it out based on what else he is eating.
Organix corn snacks (are you in the UK ? not sure if they sell them outside of here, but they are puffed corn with carrot powder on them, shaped like wotsits or cheezies. they are DS's only "treat" and he loves them and i don't feel guilty because they are just corn!)
that's really it. it is a very limiting diet to work within and it is harder with the back'n'forth bowel movements.
not sure if any of this helped or if it is just mindless rambling on my part, so will stop now!!!