Is refraining from praying in a mosque where the carpet is bad and harms the worshippers, due to the refusal of those in charge of the mosque to allow fundraising for its renovation, considered a fitna (strife/temptation) and a sin, or is it obligatory to pray in it in this condition?

It is permissible for the endower (waqif) to set conditions for his endowment (waqf), so long as they do not harm the purpose of the endowment. If the endower's condition harms the interest of the endowment, such as preventing the changing of dirty carpets that harm worshipers and cause foul odors, then this condition is not to be heeded.

Prayer in the mosque is obligatory, but the Sharia has made allowances for not attending congregational prayers in the mosque due to excuses, such as: heavy rain, extreme cold, illness, fear, and preoccupation of the mind. If the condition of the mosque is unbearable and distracts one from prayer, it is permissible to avoid praying in it, but it is not permissible to abandon congregational prayer entirely. In this case, you can pray in another mosque or pray in the same mosque on your own prayer mat.