I don't like Oh Crap generally - I think it is 99% a load of, well, crap.
I kept recognising things in it which I KNOW are total bollocks and it made me sceptical of the author's other claims which were new to me. Plus I find her peppy, know-it-all, judgemental tone absolutely infuriating and I think the insistence that you have to do it exactly her way or you've failed can fuck right off. And I think setting an alarm so you can take your sleeping-through baby to the toilet is madness. MADNESS.
But I do admit she has one extremely good tip which is to dedicate a day to watching them, she reckons naked, but I think the PJ shorts would work too - because it's just that awareness that you need to know immediately when they go - just basically observe what they do body language style when they are on the verge of having an accident.
What I learnt with DS2 doing this is that he would have those predictable signs before he actually recognised it himself. It was tiny and subtle - he would basically turn his toes inwards. DS3, I don't know. He is also nearly 3, and sounds similar to your LO. We had one perfect half day and since then he has utterly refused the idea of pants or anything and if he doesn't have a nappy on he will repeatedly cry, lie down on the floor and ask to be changed, try to climb up to the changing table where the nappies are kept etc.
We are lucky in that we have the opposite - school starts really late here (we are also abroad) and they have absolutely no pressure to be out of nappies as long as they are by the time school starts (and all the teachers/doctors insist that they will be - and even stuff like sticker charts is actually discouraged because they say it's pressure. Well, a sticker chart worked wonders for DS2.)
What we have been doing over the last few weeks for DS3 to try and get him over the upset is not insist on pants in the morning, but every time he had a bath, we'd get him undressed while the bath was running and suggest doing a "bath time toilet wee wee" and he'd usually perform for that. After a few successful times of this, DH started encouraging him to do a "bedtime wee wee" on the toilet, again much praise for this and he was happy about it. So we've added a "morning wee wee" and then occasionally he'll randomly inform us "Wee wee coming out!" so we say "Oh, well done! Shall we do a big boy wee wee on the toilet?" and often he will say yes. Nursery have told us that he has done wees on their toilet (and once, half a poo) as well.
So we are still in nappies (pull ups) but using the toilet usually 1-2 times a day but sometimes more than that. He still does lots of wees in the nappy as well. And there are times that he doesn't notice that he's going e.g. he had a fresh nappy on after swimming the other day and either a very fast flow or just a bad nappy placement meant that a damp patch appeared on his shorts anyway. He is now getting happier to be without a nappy although honestly I am not sure how he'd react if I tried to put pants or trousers on him with no nappy. I am thinking of having a day where we just conveniently "run out" of nappies unexpectedly and show him that so that he thinks there is no other option.