(((hugs))) this is hard. It sounds like she's gotten herself overtired and having trouble getting out of it. In this situation I used to always do a SHORT am nap, because I knew that if I let him sleep, he would fail to take a pm nap and then we would wind up doing bedtime insanely early. So I guess that would be my advice, to do a 45 mins am nap at 10am, and then a (we hope) longer pm nap at around 1.30. If the pm nap is short too then yes do a 5.30pm bedtime - even if she still wakes at 5.30am, at least she will have had a 12 hour night.
Walk in Walk out is as it sounds. You leave the room (even though she is crying), you stand outside the room and listen to how she is crying, if it's definitely 'I need you', you go back in, lie her down using your key sleep phrase, then walk back out again. If she's whining, shouting, taking pauses in her crying, fussing, you don't go in. Yes there is a LOT of crying involved, because sleep training toddlers is hard, hard work - they don't like change, especially when they're tired and frustrated. But no harm will come to her because you are reassuring her that you will always return if she is crying, but she has to settle herself to sleep.