No water is available in a housing unit, where one sleeps and dines, except from a sink in a room with a toilet. For purposes of N’tilat Yadaim, may one remove water from that sink, in a container, and perform N’tilat Yadaim for meals, and upon rising, in another room where there is no toilet.

The solution you suggest is permitted although the question that remains is- what will you do with the water that you used for N'tilat Yadayim? If you dispose of it in the room with the toilet, some poskim feel that you should rinse your hands elsewhere later upon finding an opportunity, since in their opinion just being in the room necessitates rinsing one's hands without a bracha. There are poskim who permit doing the act of n'tilat yadayim in a room with a flush porcelain toilet as long as the bracha is made outside of the room and when there is no better alternative this opinion can be followed.