Answer: 625

Solution:

The slant height is $\sqrt{600^2 + (200\sqrt{7})^2}$ $= \sqrt{360000 + 280000}$ $= 800.$ Cutting the cone along the ruling through the starting point and unrolling gives a sector of radius $800$ whose arc has the base circumference $2\pi \cdot 600 = 1200\pi;$ since a full circle of radius $800$ has circumference $1600\pi,$ the central angle is $\frac{3}{4} \cdot 360^\circ = 270^\circ.$ A point on the exact opposite side of the cone is halfway around, which in the unrolled sector is $135^\circ$ away.

The shortest crawl is the straight segment between the two points, at radii $125$ and $375\sqrt{2}$ with a $135^\circ$ angle between them. By the law of cosines, $$\begin{aligned} d^2 &= 125^2 + (375\sqrt{2})^2 \\ &\quad {}- 2 \cdot 125 \cdot 375\sqrt{2} \cos 135^\circ \\ &= 15625 + 281250 + 93750 \\ &= 390625. \end{aligned}$$

Thus $d = 625.$