Is it permissible for a COVID-19 patient, who fears transmitting the infection due to a shared bathroom, to pray without ablution, especially since ablution includes rinsing the mouth and sniffing water into the nose?

The basic principle is that prayer is not valid without ablution (wudu') if water is available and can be used without harm. What was mentioned does not justify abandoning ablution, as ablution is not necessarily performed in the bathroom; a sick person can perform ablution in their room using a basin. Rinsing the mouth and nose are Sunnah acts; ablution is valid even if they are omitted. Therefore, it is not permissible to abandon ablution for the reasons mentioned in the question.