Is it permissible to remove the carpet placed on the mosque's prayer mat in winter to pray on it, and what is the advice for the two parties differing on this issue?

There is no harm in praying on a prayer rug placed over the mosque's carpet, especially when there is a need for it. It has been authentically reported that the Prophet, peace and blessings be upon him, used to pray on a khumra, which is a small prayer mat. The prohibition of praying on a prayer rug applies to those who believe there is a special merit or virtue in praying on it, as opposed to other surfaces. If the removal of the prayer rug is due to its embellishments that might distract the worshipper, then this is correct, as the Sunnah calls for avoiding anything that distracts from prayer. However, if the removal is based on the belief that praying on it is an innovation per se, then they should be educated and advised to seek religious knowledge and understand the importance of unity and reconciliation of hearts.