The basic problem is that your Crimson button's action is to write to an Input point in the controller.
That's a legitimate action, and the Red Lion will do it, but the MicroLogix is going to write over it with the physical state of the Input point at the beginning of the next program scan. That's going to be just a few milliseconds later.
The traditional method of interfacing an HMI to a PLC is to read and write non-physical data table addresses. In the SLC-500 and MicroLogix families of controllers, these are typically "B" and "N" and "F" data files, for Binary and Integer and Floating Point (respectively) data.
Try re-addressing your button to "B3:0/0" and "B3:0/1", then use those in your PLC program.