The whole entry in the 1811 Dictionary of the Vulgar Tongue:

TO BRUSH. To run away. Let us buy a brush and lope; let us go away or off. To have a brush with a woman; to lie with her. To have a brush with a man; to fight with him. The cove cracked the peter and bought a brush; the fellow broke open the trunk, and then ran away.

So, 'bought a brush' is confirmed as 'ran away', and a brush with the law should be a fight with the law (officer).