V1 is a decision (action) speed - it only has to do with accelerate-stop. V1 can affect your takeoff climb but only when it affects your required VR speed, as VR cannot be less than V1.
That's not right.
Possible V1 speeds is usually a spectra between V1-Go & V1-Stop.
V1-Go being the minimum speed where you can continue the take-off following E.F. considering you now only have 1 eng to get you airborne.
V1-Stop being the maximum speed where you can still abort and stop on the runway.
The chosen V1 can be anywhere within this spectra depending if you are Go-minded or Stop-minded. E.g. if you've got a cliff or water at the end of the runway you should perhaps be more Go-minded choosing a lower V1. If you've got a big mountain in front of you it could be wise to be more Stop-minded (choosing a higher V1), as this will increase your margin in case your flying skills are less than perfect.
In practice, your company should provide you with the necessary tools. In my case we have paper RTOW charts that gives us the limiting weights for a given temperature & speeds (V1, Vr, V2). As a pilot, that is all you need to know.
For the original question, I think weight will very mildly affect Vmcg. To my understanding, the main contributors to change of Vmcg are:
- Thrust
- Pressure altitude
- Temperature