Bumping your topic because you didn't get an answer within a day is not going to make you very popular.
I don't have access to one of those scopes. The bandwidth represents the -3 dB point, so a 1 Vpp 200 MHz sine will be displayed as a 0.7 Vpp 200 MHz sine. Rule of thumb for analog signals used to be that the scope should have a bandwidth about 5x higher than the frequency of the signal. This was for a qualitative assessment on a CRT only.
For digital signals it's better to look at the rise times. For accurate rise times measurement, the system rise time of the scope and probe should be lower than the rise time of the signal. How much lower depends on the required accuracy. Here are
some appnotes on this topic.
A 370 MHz signal will most likely look like a severely attenuated sine wave, regardless of the original signal. How much attenuation depends on the frequency response of the front-end. Aliasing might also start to become a problem. Keep in mind that a square wave is composed of a large number of odd harmonics, these harmonics will be even more attenuated than the fundamental.