This is my only painting that I could find that had blue checks on it.
Painting this is a piece of cake. Completely paint the white cloth that is underneath the pattern complete with shadows, halftones and highlights.
When it is dry, mix a glaze of blue paint (I like ultramarine) and paint stripes going either horizontally or vertically (not both) over all (highlight, halftone and shadow). Be sure to respect the way the cloth moves and folds.
When this is dry, mix the same color and intensity of glaze and paint stripes in the opposite direction. Where the two stripes overlap, the square will be darker.
When this is dry, match the glazed blue color with a mixture of white and blue to exactly match what is underneath and repaint over (only) those squares (both dark and light blue) that appear in the highlighted portions of the shirt.
Because you can see through a glaze, it will be lighter and brighter in the light and show less contrast (or maybe not visible at all) in the shadows.
Viola, you're done! After this, I'll bet you ask your client to wear "easy to paint" clothes.