Sign it anywise, use a throwaway email account if you like. The idea is to match numbers with numbers. Companies like Target do things like this because they live inside a left-wing bubble and mistakenly believe that the loud noise generated by a relatively small number of pro-perversion activists is indicative of large-scale support for perversion, even though it’s not. Show them which side the numbers are really on, just like happened with Chik-Fil-A appreciation day.
One of my very conservative friends who has several very liberal acquaintances because of business issues always ones one of their email addresses. Have known him for many years and mischief is his middle name.