Ok, first of all that wheel is way too small for chinchillas, 14" is the absolute minimum diameter, 12" it will cause back injuries over time since chins are not designed to bend backwards like most other rodents. Also, it's plastic, the chins will chew it and if swallowed that can lead to a blockage in their gut which will require surgery to remove if it doesn't kill them first. Plastic is not digestible.
As to why you shouldn't allow them to breed, well lets start with, do you know their generic histories? Do any of them carry any genetic problems? or do any of their parents or grandparents? some issue can skip a generation or two. One of the worst possible genetic issues is genetic malocclusion, which has no cure and genetic malo can be the death of a chin as young as 2, do you know for a fact that none of the chins carry that? or are you ok with the possibility of producing chins that will only live a couple years before dying in horrible pain?
It's not just mummified kits, there are a lot of other issues that can arise (read
the horror stories and look at the pics, is that something you really want to get into?), like the mother could have difficulty giving birth, with people randomly breeding chins with unknown defects you do run into issues of females being unable to pass kits naturally. Did all the chins mothers produce enough milk for the kits? it can happen, requiring hand feeding. Did the mother(s) of all three have normal uncomplicated births? If you look at most breeding and show females they are large and stocky, you need a large rear end to pass kits easily, if the chin is too small or has a narrow pelvis they can have issues. If the kit(s) don't come out the mother will need an emergency c-section to remove them, otherwise they could die and start rotting, which will also kill the mother. Other mothers do need a hand to help pull the kit out, which mean having a human on hand to lend a hand. Even if the kits come out the mother can still end up with an infected uterus, or even have a kit rip something in the mother on the way out and the mother bleed to death (I've see it happen and it was horrible and sad). Chins are pregnant for so long that the kits come out fully formed (fur, teeth, nails, with eyes and ears open, and are up and about within minutes), and can be fairly large, unlike most other rodents who give birth to tiny helpless pink hairless babies.
Also do you know for sure they are not all related? if they are then you will also run into inbreeding issues. Also do you already have people lined up for kits? or are you planing to keep all the chins? they can't all just live together in one cage, the males and females will all breed, meaning brothers and sisters, mothers and sons, fathers and daughters, and you will end up with genetically messed up animals. The cage you have is barely big enough for 3 (1-2 for that size cage is better), let alone more, the less space you have the more likely fights will occur and chins can and do fight to the death.
The bottom line is breeding chins with unknown backgrounds you have no idea what you are getting genetically. You are basically just throwing more genetically unknown chins out into the world and doing more harm then good for chins as a whole. What purpose do you have for breeding? Breeding is something that should be given thought and have a purpose, like improving fur color or quality, improving size or shape, etc. It really shouldn't be just to create more unwanted chins in the world. Breeding is also not cheap, a lot can go wrong because of people randomly breeding, creating more genetic disorders. So you are more likely to run into issue with domestic chins then a wild chin, in the wild the genetic defects wouldn't survive to breed. If you don't have several thousand dollars set aside or credit cards you are better off not breeding, because you chins will likely die if they have complications and you don't treat them. Yes a lot that can happen is like humans pregnancy, but humans generally go to the hospital or at least have a midwife to help give birth. In places where women don't or can't go to a doctor, dying during childbirth is still high.
Another thing is males that breed are more likely to get hair rings, which need to be dealt with immediately. The ring of hair can cut off blood flow to the penis and if not removed quickly you'll end up having to amputate it or the chin dies.
Since you can't (or wont) not go on the trip, make sure whoever you have checking on them comes everyday, possible a couple times a day to check on them. Also make sure you leave either a credit card with the vet or a few thousand dollars just in case one of the females needs vet assistance. Also make sure the pet sitter is prepare and willing to care for and feed kits every couple hours round the clock (day and night) if the mother rejects them or isn't producing enough milk.
I guess at the end of the day they are your chins, and if you are ok with them dying because you want to breed them, and you are ok with the chance of producing short lived, weak, and/or sickly chins, I think that's cruel, but that is your choice. I don't know if you are just a little kid or what, but taking care of an animal is not just something to do when it's convenient for you. You intentionally brought these chins into your home, and you are intentionally unwilling to separate them so they don't breed and then intentionally leaving them. I'm guessing if you have to plan the trip it's not something that just came up, so you knew you would be going away.