No. As the engine warms up, the thermostat is shut, so the water in the radiator is going to be basically cold (there will be a little creep of temperature, but not much as the only access to the rad is from the bottom hose - there is no flow), hence the 30C or so you are feeling in the tank. As the coolant circulating around the barrel and head gets hotter it expands, this expansion is relieved through the bottom hose to the radiator forcing out the still cool water from the top of the radiator into the expansion tank in your case. Once the engine is up to temperature and the thermostat opens and the radiator becomes hot the coolant will essentially be at fully expanded (engine at normal operating temperature set by the thermostat and operating pressure set by the radiator cap). Now if the radiator cap fails the system will never reach operating pressure, the temperature will rise quicker than normal and the thermostat will open earlier allowing coolant into the radiator and straight out into the expansion tank for longer, and expansion will be greater, and it will do it all the time because operating pressure has not been reached and the cap is leaving the opening to the tank partially open.

You will need the correct cap, not only because it needs to release at the right pressure (16-20 psi), but the sealing surface needs to be in the right place to open and close the opening to the expansion tank. You can get them online fairly quickly from various places. Do you know anyone else who has an XT who can "lend" you their cap to try it out?

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