Is it obligatory to purify a ritually impure carpet if doing so causes hardship, given that the foot might become impure because of it and sweat, which would affect the prayer?

There is no obligation to purify an impure mattress unless one intends to pray on it. Impurity does not transfer merely by doubt, and the default state of a thing is purity. If a dry foot touches a dry impure carpet, impurity does not transfer by agreement. However, if either the foot or the carpet is wet, the transfer of impurity is a matter of scholarly disagreement. If a person is certain that impurity has transferred to a part of their body, it is obligatory to wash it. One must be careful not to indulge in obsessive doubts (waswasah), because the transfer of impurity is only definitively established by certainty.