It is not clear to me why you would not suffocate after 10 minutes divided by the number of creatures, no matter what, because the bag states:
Breathing creatures inside the bag can survive up to a number of minutes equal to 10 divided by the number of creatures (minimum 1 minute), after which time they begin to suffocate.
There is no condition on the suffocation. It does not matter if you have sources of fresh air or whatever, the bag does not say you suffocate because you used up the air.
We read that into the bag, but that is not what it says. It also would take a lot longer than 10 minutes before you use up all the are in a 64 cubeic feet space and suffocate for lack of air. A typical human needs about 450 liters of air in an hour. 64 cubic feet are over 1,800 liters. You would survive about four hours in such a space before you run out of air. So it cannot be that you suffocate because you run out of air after just 10 minutes.
If that is not it, all that remains that you suffocate after 10 minutes if you are a breathing creature because ... magic. The bag will suffocate you. Finding other ways to breathe does not really change it. You are still a breathing creature, and your 10 minutes still run out, no matter if you have a bubble of air, or if you have a neclace of adaption. You are in the bag, and you will begin to suffocate.
Water Breathing otherwise should work, too
PS. If we assume magical ways to breathe work to get around the bag's restricitons: Fill the bag partially with water, depending on your own weight, maybe 50 gallons or so, to keep under the weight limit, which will fill about half a foot at the bottom. Cast water breathing on yourself each day. Enjoy your new, wet home.
Water breathing is a ritual that does not even cost you a spell slot, runs for 24 hours and states:
This spell grants up to ten willing creatures you can see within range the ability to breathe underwater until the spell ends.
You will be able to breathe underwater for the next 24 hours, no air from the bag needed. It is however another question how habitable this is. You get kind of soggy with time.