Why can’t I go to the temple but can I go to the church during menstruation?

This statement is incorrect. You can go to temples during menstruation and perform normal religious activities such as worshiping, making offerings to Buddha, and chanting Buddha's name.

Generally, lay people go to temples to burn incense and worship Buddha. This is a kind of good deed. It is impossible to stop people from doing good deeds and cultivating blessings just because they have menstruation. Master Yin Guang, the thirteenth generation founder of the Pure Land Sect of Buddhism, once refuted the idea that women cannot worship Buddha during menstruation. The reason is the same. The original text is as follows:

Women have menstruation from the age of twelve or thirteen to forty-eight or nine years old. It is said that during menstruation, one should not pray and chant. This statement is unreasonable. If menstruation is short, it will stop in two or three days; if it is long, it will stop in six or seven days. People who practice must be mindful of it. How can they give up their practice because of a minor illness that they were born with? Now it is said that during menstruation, you can pray less (it is better to pray less, not never). You should chant Buddha and chant sutras as usual. It is advisable to change and wash dirty cloth frequently. If your hands touch dirty cloth, wash them immediately. Never turn the scriptures or burn incense with dirty hands. Buddhism, the Dharma is comprehensive, while heretics only adhere to the precipitous principles. Many people in the world only believe what the heretics say and do not know the true principles of Buddhism. Therefore, all fellow practitioners cannot benefit from the Dharma. "The Continuation of Master Yinguang's Papers·A Reply to the Letter"