I believe you when you say you are happy with Linux. It is OK. You can do with your notebook whatever you want and nobody force you to use Windows but, as I wrote before, Linux is not supported by Toshiba.
What is the DOS flash utility recommend by Toshiba for that, please ?
DOS flash utility recommended by Toshiba? Where you found info that Toshiba recommends DOS flash utility? Fact is that Toshiba doesn’t offer DOS BIOS update since, I believe, 6 or 7 years anymore.
What you can do now is to obtain second HDD. Exchange HDDs, install recovery image, update BIOS and later put back old HDD with Linux.