My philosophy is schedule über alles. As an academic, time is your most constrained resource. This goes for balancing the hours in your work vs. research, teaching, and service requirements. And in this case it goes for the limited number of class sessions in your semester.
Do you not have a day-by-day schedule for your term? You should. For me, it's the very first thing I map out when preparing a course. What you really don't want (and I've seen it happen to other instructors) is to slide loosely through the term and then realize you need to somehow squeeze 3 chapters of content into the last week of class. The awkwardness you now feel is nothing compared to that.
So keep in mind that you must proceed forward to the following areas of content. If not, be clear-headed: exactly what will get cut from the course if you give a makeup lecture? Make a deliberate choice. And what will get cut from your other work this week as you prepare the new unplanned lecture?
In a college course, students should be expected to fill in gaps outside of class on their own. Actually, that's where the majority of the work should be happening. In particular if it's your top students, and they've found an alternative method that satisfies them, at your option, just let them use either method. But for other students I would lean towards sticking with the book presentation which they can consult and reference more easily.
I recommend the Steven Krantz book How to Teach Mathematics by the AMS as the go-to for general math-teaching philosophy. Section 2.7 on "Handouts" seems relevant here:
If you give a class hour on Stoke's theorem and feel that you have not
made matters clear, then you might be inclined to draw up a handout to
help students along. You also might suspect that this extra effort on
your part will improve your teaching evaluations and, in particular,
that students will appreciate all the additional work that you have
put in. Well, it won't and they don't...
What I can do is examine my own conscience and tell you what I see. If
I give a lesson that is not up to snuff, or if I do a poor job
explaining what curvature is, or if I goof up a proof in class, then I
can salve my conscience by writing up a handout. It takes about an
hour, it is a way of doing penance, and it is a way of working past
the guilt of having screwed up in class. In my heart of hearts,
however, I know that what I should do is strive to give better
In summary: Of your listed options, (1) is the most on-target. Krantz goes on to suggest that giving a handout online as an optional added resource to students can be okay. I would recommend in the main sticking to the chosen text presentation, regardless if some advanced students have found another method. Spending more class sessions on the topic should be totally off the table, as that way lies a cascade of further problems as other topics get squeezed out of the semester.