This is going to be harsh, but you said you're concerned about your kids, so here goes.
To me it still just isn't worth it. For what you're using them for, it is an equipment solution to a skills problem. I.e. if you're far enough away from your buddy, and/or your buddy is oblivious that you can't get to them in under 20 seconds, you have skills problems. You initially claim "having a totally redundant air supply", but then state "stop-gap to get to our partner in order to share air". These are two VERY different animals. The first requires use of a 19cf bottle minimum to safely be considered a redundant air supply, 6cf won't cut it, it just won't. The second is a skills problem and you need to have better buddy awareness and buddy diving procedures. I.e. never farther than one kick cycle away.
The 13 is also not twice the weight, the cylinder is twice the weight, but as a rig, it is only an extra 3lbs, 19 is an extra 5. You still have the same valve/regulator etc. If you switch to an integrated valve/regulator, you can shave that weight off. Also, if you go to a 13 or 19, you can remove the octopus from your primary regulator set and just use that as your donating regulator which is a truly redundant air supply, which saves about a pound as well with one less second stage and hose
Dry Suit / Pony Bottle Valve "Nitrox Ready!"
If you're using jackets, you really only have cam band mounting solutions. The Dive Rite solution can be used on the cam bands, it's just not designed for it. It moves around a bit, but this gear is just not designed for use with poodle jackets.
Quick-Tec Pony Bottle Mounting System
I use this as a secure option for my argon bottle on doubles, it's not cheap, but it's secure. XS makes one that is a bit cheaper, but it is a pain when you're switching tanks on a boat since the main bottle side isn't attached until the cam band is tightened.
Also call the dive ops, they may have 30's or 40's that you can rent and sling using something like this
Cave Adventurers - Dive Rite Stage Bottle Rigging - Marianna, Florida USA - Never Undersold!
or you can mount it to the main bottle with something like this
Zeagle Pony Bottle Band Kit | Dive Right In Scuba - Plainfield, IL - Dive Right in Scuba
either way, if you really care about your kids safety, you'll get something that will actually get them up from depth and conduct a safety stop properly, and a 6cf just isn't going to cut it.
DiveNerd - Rock Bottom Calculator, Imperial Units
Play with this, you'll see that with 1 minute to resolve at bottom, 3 minute safety stop, and assuming a full 3000psi fill, you get 0 psi to play with if you can maintain a 0.6 SAC rate. Highly unlikely, especially since you may have to orally inflate if you dump too much and need to recover. You need to plan with a SAC of 1cfm for emergencies which is more realistic, especially if you aren't diving regularly, and with that, at 40ft, you have a 0psi buffer with a 6. 4 minutes total before that tank is out. Are you willing to ask your kids to blow their safety stop? Less than 3 minutes at depth before that tank is bone dry.