I believe the Buddha said whoever has awakened can only speak the Dhamma. And he asked his followers to respect any teaching that led to the cessation of suffering. (I'll link to the sutta if I can find it) He never claimed his teachings to be his own, he credited all the Buddhas with having said the same thing.
My own view is it's very simple - let's not complicate things with ideas of the dos and don'ts of a stream entrant, such lists cannot be perfect. Stream entrants too are made of dependant elements, and they will behave in subtle but different ways with respect to the time and place they live in.
Respecting other religions is natural common courtesy. Humans everywhere share the same basic qualities, the same problems and fears. Some solutions may solve some things better than others. There isn't a Buddhist blood, and a Christian blood. There cannot be therefore a Buddhist enlightenment and a Christian enlightenment. The realization of truth has to be the same. If two people get well from an illness one using modern western drugs, and another using Ayurveda or TCM, one doesn't have to be wrong for the other to be right. The same person can even use different medical traditions for different diseases. Same with religion. If the Christian Sangha is more popular locally, by all means reach out to it.
I know a lot of learned religious experts will disagree, but the truth is really simple. The qualities of compassion and kindness are the same in any human, and everyone has Buddha nature, which is not at odds with religion.
Unlike any other time in the past we are a global community today - this brings several religions and ideas together, even within Buddhism. An integration of sorts is taking place if you read the works of the modern masters like Dalai Lama and Thich Nhat Hanh, and teachers like Roshi Halifax and Shinzen Young - they want to see an integrative wisdom emerge that is not bound by a time and place in the past. The mindfulness movement of Jon Kabat-Zinn is a huge success because of its secular nature.
Just as there are so many kinds of food out there, some healthier than others, some others more affordable, some others more widely accessible, we must view religions as useful but not necessarily perfect. When I eat french fries, I know they are not healthy, and I don't like the taste of mayo on the side, but it doesn't mean I can't eat the fries once in a while with ketchup instead of mayo.
Rains are considered happy events in Buddhism, because India was a dry land, and rain was the harbinger of life. In a wet country like Ireland, rains can seem depressing. Culture and Buddhism have deeply intertwined over the years, and they need to be understood and respected.
Monks in the order of Ven. Thich Nhat Hanh are allowed to pray to Christ and the Buddha, revere St. Thomas and Avalokitesvara, read the Bible as well as the Tipitaka, there is no problem.
HH The Dalai Lama offers very practical advice in his book, Toward a True Kinship of Faiths: How the World's Religions Can Come Together, 2010
When I was growing up in Tibet, and especially after my serious engagement in studies of classical Buddhist thought and practice from the age of fifteen, I used to feel that my own Buddhist religion was the best. I thought that there simply could not be any other faith tradition that could rival the depth, sophistication, and inspirational power of Buddhism. Other religions must, at best, be “so-so.” Looking back, I feel embarrassed by my naïveté, although it was the view of an adolescent boy immersed in his own inherited religious tradition. Yes, I was vaguely aware of the existence of a great world religion called Christianity that propounds the way of salvation through the life of its savior, Jesus Christ. In fact, as a child I had heard the story of how some Christian priests had once established a mission in western Tibet in the seventeenth century. There was also a small community of Tibetan Muslims right up until modern times, who had lived in Lhasa city for over four centuries. As for Hindus and Jains, followers of the two other major religions native to India, I was convinced that the philosophical arguments, found in the classical Buddhist critiques of their tenets, had effectively demonstrated the superiority of the Buddhist faith centuries ago.
Needless to say such naïveté could be sustained only so long as I remained isolated from any real contact with the world’s other religions. [...]
Looking back to this trip in 1956, I realize that my visit to the Theosophical Society in Chennai (then Madras) left a powerful impression. There I was first directly exposed to people, and to a movement, that attempted to bring together the wisdom of the world’s spiritual traditions as well as science. I felt among the members a sense of tremendous openness to the world’s great religions and a genuine embracing of pluralism. When I returned to Tibet in 1957, after more than three months in what was a most amazing country for a young Tibetan monk, I was a changed man. I could no longer live in the comfort of an exclusivist standpoint that takes Buddhism to be the only true religion. When tragic political circumstances in 1959 forced me into exile in India to live as a refugee, I was paradoxically afforded the freedom to deepen my personal journey of understanding and engagement with the world’s faith traditions.