We will assume a flat earth here, as on the true earth the actual distance from the base camp to Lake B will be a little bit greater due to the curvature of the earth.
Write the vector from the base camp to Lake A as one of magnitude 360 and pointing at 20 deg but using x,y
This is 360*cosine(20) i + 360*sin(20) j or
338.2893434829270182594793398 i + 123.1272515972407438958758613j
Then write the vector from Lake A to Lake B which is magnitude 200 at 30 deg W of N which is 30+90 = 120 deg. This vector is
200*cosine(120) i + 200*sin(120) j or
-100i + 173.2050807568877293527446341j
Then you add the two vectors to get the vector from base camp to Lake B
238.2893434829270182594793398i + 296.3323323541284732486204954j
The magnitude is 380.2560484935933662279559516 km and the direction from Lake B to base camp is 180 deg opposite of our vector here.
The direction of this vector is 38.80376136571472017699068037 deg N of East, so the direction back is 38.80376136571472017699068037 deg S of West.
You can round this all to 380.256 km at 38.8 deg South of West
