You can also just dump your rootfs in /data. Since Note 3 is datamedia-based it has a huge ext3 data partition shared for apps and user data. You can take advantage of this by sticking your chroot there and having access to shared space. Then just write a script to chroot into /data/debian or whatever distro you use. I've been using this for a few months with great success.
Samsung Galaxy Note 3 SM-900T 32GB
Samsung Galaxy Note SGH-i717 16GB
HP TouchPad 32GB
Archos 43 Internet Tablet - 16GB + 32GB mSD
Dell Axim X50v - 128MB + 16GB CF