The OP says father's brother's wife and how can it be a different கோத்ரம்.
Opinion to OP. If you feel you have to follow the rituals then you should follow to not to visit temples for one year till varushapthikam because she is your gothra and for gothra தாயாதி one year is banned to visit temples celebration of festivals at home and even the kolam at entrance
Birth Gotra is hidden that's all. When blind believes do not change without knowing the intentions, fear and blind believes take over, it destroys the religion. Religion changes with respect to time but the intentions do not change. Hindu religion is not static, it is dynamic and that is why it is surviving. Rigid system collapses. Take for an example, Sanskrit, starting from grammar to learn it was the old method and it died. If we have adopted to speak first – that is what a child does - and then introduce the writing and then grammar, it will be alive to day. The seven cases in Sanskrit are similar to the seven cases in German. They are fine for literature but not for spoken form. If one can manage with 800 words of English, why not with Sanskrit. But none did the survey and research to do it. All reforms were blocked, with the result it is almost dying. People who are working in offices,s schools etc., do sandiyavandanam three times a day, homam, agnihotram? Yet still the person claims to be a Brahman is a myth. When girl get married she still caries her parental DNA and that does not change after marriage because Gotra of the Husband is different. She donates her genetic material and if is the same like that of her husband, it will not be a child! How come her Gotram suddenly becomes that of her husband, if so is she marrying in her own Gotra?
Mourning is a ritual and prayers for the departed soul and must be respected. But one can still carry on with one's life. Even in a death house, people still drink coffee, eat lunch etc. Even the vadiyars do not do Sandhiya vandnam when they are traveling! Do they say like historical time, let me finish Madyanigam etc.? What is important is mourning and making life miserable at the current situation - modern times working 8 -5 etc., needs adjustments. No where the religion says that you should be rigid.
When you tell the Muhurta Mantra that we will be like two oxen pulling a cart by putting the nookaal, on her neck and pledge that you will be moving in the same direction (do every thing)are we doing it really? Even do you treat your wife like a real Ardhangini or control her.?
The person who asked the question has to think and do what he wants. GODs never punish him and such GODS will not be GODS will be just ordinary human being.
Sorry for sharing my view. All right it serve no purpose arguing here.