In the ancient times, marriage meant 3 things. Kanya daan, paani grahan and saptapadi. Kanya daan where a pre-puberty girl is promised to another family is not applicable now.
Pani grahan is applicable in which, the girl, whoever she is, from whichever family (or caste) becomes part of a new family. Her name, caste everything is new. The groom holds the hands of the bride and makes her part of his family.
Then we have saptapadi where 2 equal friends promise to each other to be together like sky and earth, word and meaning etc etc..Actually in all 8 types of marriages, saptapadi is the actual marriage ceremony.
If the girl who comes in becomes part of your family and is willing to perform all the karmas and anushtanams, then I do not see any issues in the girl taking part in every anustanam or nitya karma of yours. In the name of her birth being in another caste, rejecting her is anti-sastra in my view, though several people may disagee with me. The reason it is anti-sastric is because once you hold her hand and accept her, she is your family. Nothing else. So she has all the rights to participate in every karma and anushtana.
If she is not willing to do that, then you can still continue to do that on your own. Without wife's permission, you may not be able to do some rituals like aubasanam (with fire), but pretty much you can do everything.
Any marriage intercaste or intra-caste does not depend on these rituals. As the vedic mantras say man and woman are different beings, like sky and earth, with different natures, expectations etc. At first they are held together by sexual attraction (a la gravity). As it waxes and wanes, the societal pressures (children, parents, friends, relatives) hold them together and they evolve through a period of turbulences and fights. As they grow, learn and mature, their individual bond becomes stronger. But for a good amount of time in between, it's the social pressures that work on them, not to diverge.
In the past, inter-caste marriages do not give that social pressure, while intra-caste marriages exerted that pressure. Hence they were looked down upon. In the modern world, it's no longer so. Social pressures have lesser and lesser value these days.
Hence it boils down to individuals and how much they are willing to walk through this family fire, that melts them and fuses them. Whomever you marry or live-in, if you go through this, with give and take, with sacrifices for each other, accepting each other as individuals with their positives and negatives, then you will get through this and have a long lasting relationship.
So instead of worrying about rituals, talk to your partner about how you people want to live your life together, how you are going to face the changes in each other as you age etc etc..