How do I correctly apply KVL in this situation, and why?
To apply KVL, add the load between v1 and ground to your schematic. Then make a directed sum of the potential differences; if I follow your direction of I, I get:

+0.7 - 3 - v1 = 0

That's because the direction over the load would go upwards to get back from your GND to the point labeled v1 in your schematic.