Contact precautions and nowhere to wash my hands?

Our hospital policy states you must have gloves and gown prior to entering a patient's room that is in contact precautions. Fair enough. They also state that we must wash our hands using soap and water after removing our gloves. Again, completely reasonable, I'm in agreement with that.

The rooms at the hospital I work at are pretty small. The sink is inside the room, next to the patient's bed. It is a shared sink, patients use them to wash their hands too. To get to the sinks you have to walk into the room. The next nearest handwashing station is at the nurse's station.

To wash my hands with soap and water, I have to enter the patient's room after degloving and degowning, which I'm not supposed to do (per policy and if I was seen doing it I'd get dinged by the infection control nurse). Otherwise, I have to go to the nurse's station and handle the sink handles there and hope I don't touch anything accidentally on my way. Not to mention, if I'm touching the sink handle at the nurse's station I'm contaminating it.

So, my question is, in your opinion, what is best practice in this situation?