Bottom line is 'yes' you can dye poly cotton in a washing machine, just as you would any other fabric. The down side is that it may not be the same colour throughout. So you have to decide if you like beige better than an uneven alternative colour. The problem is that cotton dyes darker than the 'poly' part.
Yes it shouldn't be problem. I had these poly cotton socks that my daughter decided she wanted to dye and they came out just fine.