IIRC Dragonetti and Bottesini's dates barely overlap (April 7, 1763 April 16, 1846 for Dragonetti and 22 December 1821 7 July 1889 for Bottesini according to wikipedia). Bottesini didn't visit England until 1849, after Dragonetti had been dead for three years. While he had had some success as a soloist in America (most notably, in Cuba, where he was also the music director of the Italian Opera), the amount to which Dragonetti would have known or cared about this 20-some year old man when he was in his late eighties is definitely up for debate.
As for Dragonetti being old school, well, of all instrumentalists I've met, string players and bass players in particular are the least accepting of change, so it wouldn't surprise me in the least if Dragonetti looked at an overhand bass bow (such as the one in Bottesini's photo on wikipedia, and quite a club in its own right) and dismissed it as a foolish idea.
They both agreed that a 3 string bass was the way to go, however. They did tune it differently (ADG or higher scordatura for Bottesini and GDG or ADG depending on the occasion for Dragonetti) but both preferred the lesser tension on the top for optimum sound.
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