Grub4dos is not going to do an install of Bionicpup.
Grub4dos is a boot loader used to boot the computer and the operating systems on it.
If you have done this as you indicate you did.
Turn off your device, open the back and remove the Write Protect Screw.
Press Esc + Refresh + Power
Press Ctrl + D and press Enter to enable developer mode
Press Ctrl + D to boot Chrome OS
Open a tab and press Ctrl + Alt + T
Type in "shell" and press Enter.
Paste the following into the shell and press enter
cd ~; curl -L -O https://mrchromebox.tech/uefi-flash.sh && sudo bash uefi-flash.sh
The Chromebook now has a mrchromebox uefi firmware on it. So the bios firmware acts like a normal computer.
You are going to need to use a Puppy installer that can also install a UEFI boot loader to boot it.
I suggest this:
Do all of this from Bionicpup running from the Bionicpup USB.
Use Gparted program
Note:
Not knowing exactly what partition table setup the drive has.
You could keep it the same or change it to msdos.
Then do partitioning.
Delete all partitions on drive and start over.
Setup the internal drive with 2 partitions.
First one, small 300MB, fat32 format, flagged boot. (location for boot loader files, boot partition)
Rest of drive, whatever other partition(s), but one ext3 or 4 format. (location to put frugal installs)
(this is UEFI standard requirement, and some computers, look for a fat32 partition, for boot loader files)
Use Frugalpup Installer to do the install.
Run Frugalpup Installer main program.
On the main window are selection buttons.
Select the Puppy button, to do the install.
Go through install process, selecting to install to the ext formatted partition.
Note:
When selecting the partition to install to.
A window pops up, giving option to make a directory, to put the frugal install in.
I make this directory and usually name it, the name of the Puppy version.
Carefully read that windows info.
Press enter, makes the directory, not the OK button.
complete the install.
When it gets back to the main Frugalpup window.
Select the boot button.
Select the location of the frugal install, on the internal drive.
Select the small 300MB partition on the internal drive, as location to install the boot loader. (may need to scroll the selection window)
Select the boot loader type.
UEFI
mbr ->legacy bios boot
both
You can install the UEFI for UEFI computers, mbr for legacy bios computers, or both, to boot anything.
You probably need the UEFI one.
Now you should have Bionicpup as a frugal install and the boot loader installed to boot it.
Shutdown poweroff the computer.
Remove the Bionicpup USB.
Now you have to get the computers bios setup to first try booting from the internal drive.
Poweron and boot the computer.
Just as it starts should see a screen where you can press a specific key to enter the bios setup.
Do that and go to the config where you can change the boot device order.
Make the first boot device the internal drive.
Save change.
If this is the same mrchromebox firmware I have. back at the main config screen you can select use default boot.
It should boot to the boot menu with selection to choose to boot Bionicpup.
To put more frugal installs on the same drive.
Do the complete Frugalpup Installer process again, for the new Puppy version.
When you run the boot loader install.
It will make entries, for all installs it finds, on the drive.