You can't. First, here's a lemma:

Given three distinct lattice points (points with integer coordinates) A,B,C in the plane, either ⦟ABC is a right angle or tan ⦟ABC is rational.

Proof: Assume that ⦟ABC is not a right angle. If either line AB or BC is vertical, rotate the plane by 90°. Then at least one line is horizontal, and the other can't be vertical, because ⦟ABC is not a right angle.

Let x and y be the angles AB and BC make with a horizontal line. Then ⦟ABC = |x - y|. The tangents of x and y are rational, because the points have integer coordinates. Then |tan ⦟ABC| = |(tan x - tan y) / (1 + tan x tan y)|, which is rational.

The hat has angles of 120°, and tan 120° = -√3, which is irrational, so you can't draw the hat with lattice points, no matter how you scale or rotate it.