John - here's my 2 cents on your question. First off, I am very familar with Mammoth having grown up skiing there; now I pretty much only ski Tahoe (Alpine Meadows & Pali (squaw)). I own the AR, have skied with AX quite a bit, and only skied the SR88 once.
Between the 3, it's a sliding scale tradeoff between on and off piste. The SR88 is the best "all around" and is likely a tad more forgiving....a great generalist that competes with a slew of good all (front side) mountain skis on the market (ex. Enforcer 88, Brahma) albeit with the classic Stöckli feel and look; for those off piste moments this ski will do well and still can carve up the mountain. Between the AR vs AX, I spent nearly a year debating the virtues of these two skis; both fantastic and both faster more carving oriented vs the SR. The AR is faster, more demanding (ie, requires a higher skill to get the best out of the ski but you feel that good energy back too), and IMO it slices through chop and soft off piste areas with more leadership. BUT, because it's stiffer, the AR will be inferior to the AX in tight trees or bumps. The AX has a big following for a reason, it's more approachable yet still very fast and carves amazing when pushed. Last year to this year, the skis are the same other than cosmetic.
I'm an advanced, slightly aggressive skier. I demo'ed the 175 in all skis. I'm 5'7, 140lbs, and after long debate ended up staying conservative on sizing and own the AR in 168; however, I would have very likely done 175 in AX (since I thought this ski was less demanding and I would only use it on piste - I have other skis for off piste - my AR is my no speed limit hot lap ski and it really excels at that!!).
It sounds like your sons will be best on one of each 168 for the lighter kid and 175 for the bigger. For On Piste nothing will be more rewarding in Mammoth than either the AR or AX; but indeed, it could be that an SR could be a better fit IF, big IF, you all end up off piste (or if they do more and more with friends, etc). There are alot of good skis on the market. It's all personal preference. If I only had one SKi, while I LOVE my AR, it just doesn't work as well as my proper all mountain skis in powder, tight hop turn situations, or when in the air (all of which I strive to find); but I LOVE the AR for carving the corduroy. I would not compare the SR to the AR/AX; rather decide which ski group is a better fit then go down that path; and if they don't love to push the ski hard to feel it flex and feel your quads hurt, than perhaps the AX is a better fit. Again, I love my AR, but I am aggressive and love to push it. Good luck