Solution:

Let $\angle A = \angle C = \alpha,$ $AD = x,$ and $BC = y.$ Applying the Law of Cosines to diagonal $BD$ in triangles $ABD$ and $CDB,$ $$BD^2 = x^2 + 180^2 - 2 \cdot 180x\cos\alpha = y^2 + 180^2 - 2 \cdot 180y\cos\alpha.$$

Rearranging gives $x^2 - y^2 = 2 \cdot 180(x - y)\cos\alpha,$ and since $x \ne y$ we may divide by $x - y:$ $$\cos\alpha = \frac{x + y}{360} = \frac{640 - 2 \cdot 180}{360} = \frac{280}{360} = \frac{7}{9}.$$

Then $1000\cos A = \frac{7000}{9} = 777.7\ldots,$ so $\lfloor 1000\cos A \rfloor = 777.$