After reading some of the ISY-26 user’s guide, you definitely do not want the ISY in “linking mode” at the same time as you are trying to link a SwitchLinc to the EZIO6I. While the ISY is in linking mode, actually multi-link mode, it is listening for multiple responders. As I mentioned before, the EZIO6I is not a responder. The result of having the ISY listening for responders at the same time the EZIO6I is in link mode listening for responders is not predictable. The ISY users guide discusses a ControLinc option for linking a ControLinc to the ISY. They recognize that the ControLinc is a special case because it is not a responder. You might be able to link an EZIO6I to the ISY using the ControLinc option but the device type will not match that of a ControLinc so the ISY might not accept it. Also the EZIO6I requires that each Input have a group number assigned to it. This is taken care of automatically when a responder is linked to an EZIO6I Input using the Set button tap procedure. I don’t think the group numbers will be assigned using the ISY ControLinc link option. The SHN Utility Suite provides a means for assigning the group numbers to the Inputs of the EZIO6I but that is not an option in your case.
EDIT: The 4 second Set button press sequence that ISY support suggested to link the EZIO6I is close (although not exact) to the link sequence you would use if linking an EZIO2X4 to the ISY. The EZIO2X4 is a combination device, a controller and a responder. Recommending the use of the responder link sequence suggests the ISY support folks were not aware the EZIO6I is not a responder.