I sympathize with Axler's "Down with Determinants!" because when I was originally taught determinants the definition was offered without any motivation. Later, in an upper division course, I learned that a determinant could be interpreted as the product of the eigenvalues, and it was only at that point that I felt that I understood what the determinant was. This is the definition that Axler proposes using.
There is nothing wrong with this approach in principle, but it only makes sense when you're doing linear algebra over the complex numbers. That means that it's incompatible with your chosen text, since Strang does determinants fairly early and gets into complex numbers only at the very end of the book. I don't think it's a good idea to depart so radically from the sequence of topics in your chosen book. You describe two courses, a standard sophomore review class and a 5-week rapid review for students who have already had that class. For the standard intro class, I would predict massive failure. Most of your students will be engineering majors, and although you're at a school with highly selective admissions, I think you'd find many students really struggling to figure out what was going on if you departed this much from the presentation in the text. For the review class, you might be able to get away with more, because they've already seen the subject, but the rapid pace of a 5-week class also leaves less room for error.
Axler's main complaint, echoed by my experience, was the lack of proper motivation for the determinant. I looked at the motivation for this topic in the fourth edition of Strang, and IMO it is indeed pretty bad. He introduces the determinant casually in problem set 2.3, #8, with essentially no motivation (or none that a real-world student would be likely to pick up on). Then he never mentions determinants again until ch. 5, where he abruptly starts talking about them in a tone that suggests that students already know all about them.
My suggestion would be either to use a better book, or just to use your lectures to patch up these defects in Strang. As an alternative book, Hefferon is really nice, it's free, and the motivation for the determinant seems much more acceptable to me than what Strang does.
If I was constrained to teach from Strang or wanted to use it despite the bad presentation of the determinant, then I think my "patch" presentation would go something like the following.
We've seen that a matrix can be interpreted as a kind of transformation, a function that takes a point in R^n to some other point in R^n. As with many things in life, these operations can be undoable or not undoable. If you make a typo in your English paper, you use the undo function in your word processor. But if you launch the nuclear missiles, there is no undo.
Unfortunately, it's not always obvious from looking at a matrix whether it's invertible or not. This is mainly because a matrix can be expressed in more than one basis. For example, diag(1,0) is obviously not undoable -- geometrically, it's a projection onto the x axis. But if we rotate our basis vectors by 45 degrees, we get this matrix ..., and in this form it's not at all obvious that the matrix isn't invertible.
A nice way to see what's going on here is to take a unit square in the Cartesian plane and see what happens to it when we apply the matrix. Diag(1,0) takes this square and squashes it flat, making its area become zero. Clearly if you eliminate the area of a geometrical figure, you're losing some information, so that's not an invertible operation. Every linear transformation in R^2 has some number associated with it, which is the factor by which it changes areas. That number is called the determinant. Here are some other examples, interpreted geometrically. (Show a rotation and diag(1,2) as examples.) You can see that for a diagonal 2x2 matrix with positive elements, the determinant is the product of the diagonal elements.
Our definitions will all actually come out simpler if we reinterpret "area" to be a signed quantity. You've already seen this kind of redefinition in high school geometry and trig, where at some point you learn about negative angles. With this redefinition, the determinant of a diagonal 2x2 matrix is always the product of its diagonal elements. When the determinant is negative, we're mirror-imaging things.
Two sweet things about this definition: (1) It's independent of what basis we choose. (2) It's actually not hard to generalize to R^n. In R^1 we would just change "area" to "length", in R^3 we'd use volume, and so on.