My advice would be to teach the traditional course. Especially because you lack knowledge of ODEs themselves. Deciding that you are going to do something different when you lack knowledge, experience of the topic and likely sympathy with the objectives of the course (teaching engineers).

Rota has a fun to read essay and there are parts of it that I like. For example 2nd order ODE with constant coefficients is very common in science and engineering. however, even here the course really already covers it heavily. In some cases repetitively: calculus primer on ODEs, ODE itself, and then in majors courses in physics, chemistry, engineering. I would also emphasize that Rota's book never had great commercial success either as a common text like Brown or an underground Amazon success like Tenenbaum. Read the reviews of the Rota text on Amazon for some perspective.

Once you are a little more experienced perhaps it makes sense to do something revisionist but I would be very hesitant of the "I'm a number theorist and will teach all the engineers their ODEs differently since it was all wrong before...even though I don't even know much about the topic or about teaching it"