Since you asked for different approaches (even anecdotes), so giving mine. (My practical solution is at the end.)
My undergrad school had a similar engine math course. It was semester-long and used Kreyszig as the text. Which is a quite good book, imo, with sound explication and drill, but also wide coverage. Of course the problem is that it would take 4-6 semesters to cover it all! And the students don't have time for that. They have to take statics and thermo and etc. To the extent that it already constrains their humanities electives more than non-engineers (chemists for example). So, the solution is to prioritize. They get a deeper exposure to PDEs (over what the diffyqs course gave) so they've at least dealt with "Jo and Yo" (Bessel functions, J(0) and Y(0) and the vibrating drum problems). And a few weeks on linear algebra (which is basic matrix manipulations). This, because they DON'T have time for a full semester class on LA. (Something math majors are often shocked to learn, but is common.) Complex analysis was NOT covered at all, despite being in the book. Really, only a small amount of the book was covered.
When I went there, the solution for mathies was that math majors NEVER took engine math. They took the same classes through diffyqs (calc 1-3 plus diffyscrews (almost entirely ODEs). But never took engine math. They moved into real analysis (called "theoretical calculus") and abstract algebra and the like. They also had a required full semester in linear algebra, along with the option for additional courses in that topic. PDEs and complex analysis were electives and full semester courses, meaning some of the students never took them, although realistically, the majority did...and if someone prefers taking an operations research course to PDEs, we should realize that "different strokes for different folks". There is not time in the undergrad to take everything, hence why we have majors to start with!
The solution for physics was to take engine math...and then any special topics were done in physics class (e.g. learning calculus of variations in the second mechanics course). This is non-ideal, but life is a constrained by time problem. A few years later, they just felt this was inadequate and installed a two semester math prep course. So essentially physics (and EEs) got a second semester of engine math that covered the complex analysis (of Kreyszig).
So, practical alternatives: diverge the math students out of an engine math course (they never even take it). And for physicists and EEs give them a second semester (and figure out what to cover in it, but it's mostly an engineering-oriented complex analysis course. Maybe a little more on PDEs also. To be practical, I would just do that as an appended course. Not attempting to "enrich" the previous semester, while running or integrate into it. [Note that this is not unreasonable. ODEs is treated again after a short exposure in second semester calculus. And it is normal for physics, chemistry, and biology to be treated at a college level after an earlier high school level exposure. We routinely get questions on this site for "why don't they do it right from the start", but this ignores practical pedagogy and the limits of human comprehension, as well as the option value of those who take a simple course and decide later to go deeper.]