So there are obviously different ways of playing this.
As a general framing, lots of people start in Engineering and then switch out for various reasons, indeed it can be a majority at colleges where there are good alternatives. So I am always hesitant to endorse a college which will only really make sense for you if you actually stick with Engineering, unless perhaps that is your only financially viable path into Engineering.
On the other hand, as people are pointing out, starting at a college with no Engineering can be problematic even if you can potentially do some sort of 5-year plan, including because of the possible additional cost (including opportunity cost, which can be no small thing, but also you have to watch how aid would work and so on).
So I think ideally, prospective Engineering students should choose a college which is comfortably affordable, has a good Engineering program (usually ABET certified programs are good and then possibly a few others) if they choose to go that way, but also is a good place for other majors if they end up not wanting to do Engineering.
And there are plenty of colleges like that, but it sounds like your application strategy did not lead there. So now you have to do something less than ideal, and I think you need to think carefully how you want to handle it as different people might reasonably handle it differently.
So, one plan is obviously stick with Pomona and just deal with the possible costs of a five-year Engineering plan if you end up still wanting that. The “good” news is you probably won’t! Most people who start at a college like Pomona with that thought end up just not doing Engineering. But a few do.
Another is find an affordable ABET Engineering program that will still accept you for fall enrollment. As I see it, the main downside is whether or not that will really be the right college for you if you end up not wanting to do Engineering.
And a third is to take a gap year and restart this whole process with a new focus on making sure you end up with some suitable college choices. But that ALSO has an opportunity cost of course.
As a final thought, if it was me I would just go to Pomona. But then again, I have never wanted to be an Engineer, and so it is sort of hard for me to really imagine caring that much that not a lot of people starting at Pomona will end up Engineers.
Still, you have a great college offer which is affordable for fall admissions, so that is a very tempting way to start. And then you can figure out the rest in the future depending on how things actually evolve.