Step 1: Understand the Speeds
Let walking speed be $v_w = 5k$ and running speed be $v_r = 1.4 \times 5k = 7k$.
Time taken is inversely proportional to speed for a constant distance: $T \propto \frac{1}{v}$.

Step 2: Use given time data
Given: Time to walk $B \to C = 3$ hrs $30$ mins $= 210$ minutes.
Given: Time to run $B \to A = 3$ hrs $30$ mins $= 210$ minutes.

Step 3: Calculate required segment times
To walk $A \to B$: Since running $BA$ takes $210$ min, walking $AB$ will take $210 \times \frac{7}{5} = 294$ minutes.
To run $B \to C$: Since walking $BC$ takes $210$ min, running $BC$ will take $210 \times \frac{5}{7} = 150$ minutes.

Step 4: Final Calculation
Total time $= \text{Walk}(AB) + \text{Run}(BC)$
Total time $= 294 + 150 = \mathbf{444}$ minutes.