Its not insulting. Its your exact story. You claim that your working class grandmother was impregnated by the unmarried Nicholas II when he was Tsetsarevich. You have been told repeatedly that Nicholas was thousands of miles AWAY from England during the period of impregnation. Despite this, you claim that Nicholas was SOOOOOO smitten with your unmarried grandmother that he dashed secretly to England to have sex with her (she being unmarried and of much lower social status than a Grand Duke of Russia makes her a tart, whether you like it or not...it does.)
As your grandfather was born out of wedlock he is by definition a bastard. Sorry if the truth about your family is insulting, but this is what YOU have told us.
As for the DNA analysis on the Teinment tooth, Dr. Melton at Mytotyping Technologies will I'm sure be DELIGHTEd to do the mtDNA. As Teinmet claimed to be Alexei, and Alexei was the daughter of Alexandra, his mtDNA will be that of the Queen Victoria maternal direct descent line, which she already has sequenced and has in her files courtesy of HH The Duke of Edinburg. The Teinmet tooth is a no brainer, so get hopping on it.