Is what is done by one who invokes Allah's name at the beginning of eating, then repeats the invocation several times during it, and resumes eating to invoke Allah's name with another morsel after finishing, Islamically correct?

Do not repeat the Tasmiyah (saying "Bismillah" - In the name of Allah) when eating if those present inform you that you have already said it, and beware of obsessive doubts (waswas). If you predominantly believe that you have said the Tasmiyah, then do not repeat it. If you have not said it, then the Tasmiyah of one of those present is sufficient for the rest, according to many scholars, and this is what Al-Nawawi and Ibn 'Uthaymeen preferred.