Thanks, rcrsn51, for responding. I had an idea which I thought wouldn't work, but I couldn't remember the name of the 'ingredient' and procedure it was missing: mbr (Master-boot-Record).
The idea was --essentially-- to use the files on the USB-Key of his remastered Bionicpup32, including those he had to install to get it to boot as the source for a new ISO perhaps using ISObooter to create an ISO from the source. But I don't think that copying files from the USB-Key would capture the mbr for inclusion as a component of the source.
Giles, you have a rather unique problem, complicated by a number of unknowns.
For an application to be fully functional, it must fully interface with the operating system it runs under; and that operating system must fully communicate with the hardware required to perform its functions of and on the computer it is being run on; Other than for UEFI computers, I'm unaware of any instance in which legacy-grub could create a boot-able system when grub4dos couldn't. One unknown is why that happened on your computer*,
Are you sure the problem will materialize on your friend's computer?
Other unknowns are what operating system your friend will be using to create the Bionicpup32 on a USB-Stick? What programs or applications for creating that a 'USb-Puppy' does that operating system support? Does his/her computer even support the 'out-of-the-box' Bionicpup32?
What a remaster does is modify the 'Base' --core components-- of a Puppy operating system: removing some User applications, adding some, updating some. By 'User' applications I mean the applications a User knowingly interacts with: The Window-manager, the File-Manager and the applications a User opens to accomplish tasks such as browsing the web or writing a document. Remastering does not change the kernel or the drivers and firmware used by the kernel. Nor does it change those files provided in the original ISO to create a boot-loader and its configuration files.
Giles, Bionicpup's application to create a boot-loader includes alternates: components for UEFI computers and those not using that system. Run on the later, the UEFI components are ignored. They should not have interfered with booting on your system.
Perhaps there is some easier means by which your friend can obtain a functioning version of the remastered base you built?
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* A problem which arose a couple of time is that prior formatting of a USB-Key or partition interfered with grub4dos' writing the mbr.