Problem #1: To start, is the radiator in the hall bathroom connected "in series" or "on the same run" to the cold two upstairs rooms? If so, you may need to circulate the water through the system faster, or may have a clog. If the three radiators are "in series" and two are cold and one is hot, you need more water flow through the rads.
Do you have a multiple zone system? If so, the zones to the cold rooms are not working properly.
If you had a leak in the system to the extent that you describe, the system would run dry in a few hours. You may have air in your system, and it the cold rooms have the highest radiators and/or are last in the series, they could be full of air which would produce little if any heat. However, I am sure that you have blead any air out of the system as the very first step.
How would filling the tub from the sink help??? Its the same volume of water you are trying to fill into the tub???
Brian Woods may respond with tankless answers. However, without more information, I'm just guessing???