You can do it from a live cd/usb just fine, but you’ll need to repeat some preparation steps (basically chapter 2 of the book) every time you boot the host.
The way I did was to install the host (I chose Debian) on a small partition on the disk, leaving most of the space for the LFS partition. Once the (B)LFS build was complete enough that I didn’t need the host anymore, I just removed its partition and expanded the LFS partition into the now unallocated space.
You can absolutely do that, but I would highly recommend setting up the laptop for SSH access. That way, you can work on your LFS from your main desktop, which will probably be much more comfortable, will allow you to easily copy and paste from the book and so forth.