You can Attack and Ready an attack this way
But you might not need to.
Based on the fact that you mentioned the Crossbow Expert feat, and the "counted as you attacking on your turn" bit, I suspect you are concerned about the Loading property of Crossbows. Loading is sometimes simplified to "You can only use it once per turn", but the actual text listed under Weapon Properties is as follows:
Loading. Because of the time required to load this weapon, you can fire only one piece of ammunition from it when you use an action, bonus action, or reaction to fire it, regardless of the number of attacks you can normally make.
In other words, if you use your action to Attack with a crossbow, you can Action Surge and Attack again with the same crossbow because you would still only fire the crossbow once per action. Using the Action Surge to Ready an attack also works, but it isn't necessary if you just want to attack with the crossbow twice.