Did you compress the caliper piston when you removed it to give it some play?
I can't picture that exact caliper right now, but I've seen plenty of calipers that anchor the pad at one end and the other just rides on a guide.
Think of it this way, if there was a way for it to pop out, there's room for it to seat again.
But if you didn't compress the caliper piston before, go ahead and do it now carefully by hand, or with a small prytool and maybea thin plate or something against the brake pad so you're not pushing directly on the pad.
Once you've got a little play, position the pad again.
Worse case, pull the pin and reposition the pad then put the pin back in.
Not sure why you had so much trouble aligning the wheel, that's actually pretty easy. My guess is that you were overthinking it or trying to hard. If a single tapered pin isn't enough, use a pair of them get them into individual lug holes and gently move the wheel back and forth until you've got it aligned.
Most of all, remain calm in these situations, getting frustrated only makes it harder.
Hope something there helps.