Like you said, V1 is selected from within range of speeds. Upper end = maximum speed where stopping is still possible & the lower end = minimum speed where a continued take-off is still possible. WAT, TODA & SDA affect those limits, reducing the gap to zero as they converge to a balanced field length.
Continuing the take-off (from a lower V1) is mutually exclusive to stopping (from the higher V1). Once you continue through Vr you're no longer in a situation covered by performance data for stopping - so a higher V1(stop) can't apply. Once you start stopping then you're no longer in a situation covered by performance data for continuing the take-off - so a lower V1(go) can't apply either. End result: a single V1.
No doubt it would be possible to have a range of V1s for each take-off, each for various criteria, but the continue & stopping performance can be so
critical that taking time to decide which applies & then acting would lead to an unacceptable risk. A significant point about using a single V1 is that it simplifies decisions at a critical time ie above you go, below you stop.