When did LTI systems become part of the EE curriculum?
So


  when did it become part of the standard curriculum


can only be defined by finding those who defined standard curriculum, and catching them the moment they establish that standard.

Well: 

Welcome to Karlsruhe, home of the former Universität Karlsruhe (Technische Hochschule) [German, roughly: University of Karlsruhe (Institute of Technology)]; place where Heinrich Hertz proved the existence of the electromagnetic wave as propagating physical entity based on Maxwell's differential equations.

The first EE degree/course started winter semester 1885/86 (homepage of the faculty), and Hertz was teaching there right from the start; so, here's the first curriculum:



See "Dritter Kurs" (third course, i.e. third year/sixth semester)

Seeing that the poor students were exposed to Hertz's definitely physical/differential analysis focus for more than a year worth of courses, we can be pretty sure they learned a lot about systems that are linear, time invariant and overall pretty close to what we know about these systems nowadays.

Now, the shape in which Hertz handled Maxwell might not have been the same as it is today, and of course he couldn't rely on a rich history of control theory with description of optimal control strategies etc, but it's certain that these students received what would become part of the standard curriculum a couple years later at many universities, especially in the German-influenced room (which, arguably, was very influential in the world of late 19th/early 20th century electrical engineering, not to a small part due to Hertz's groundbreaking discoveries). 

Now, having checked that the first EE course (which was founded just three years earlier, 1883) at TU Darmstadt (which is not that far away) didn't have that much theory on its curriculum and was more involved with very practical aspects such as electrical lightning and making electrical machines rotate at all, I think we might have traced this down.