Is it permissible for someone afflicted with a rare type of allergy that causes him to choke when water touches his fingers to perform Wudu without washing his feet, based on the doctor's advice that contact with water worsens his condition and interacts with the prescribed medication?

If a trustworthy doctor informs you that washing the feet with water will cause an illness, then it is permissible for you to perform tayammum (definition) instead of washing them. So, you wash the rest of the ablutionary parts, then perform tayammum when you reach the washing of the feet.