The solution is to get a reference to the page ScriptManager then use the method RegisterPostBackControl on your postback control. You get a reference to the page ScriptManager with the static ScriptManager.GetCurrent(Page); method. You get the control you need inside the templated control Item/RowCreated event with a e.Item/Row.FindControl("postbackControlID");
So, the end result is:
Of course, if you want it the other way around (set the controls as Ajax async postback triggers) use the RegisterAsyncPostBackControl method instead.
Special thanks to Sim Singh from India for asking me to research this.