Changing anything can bring on a bout of frustrated crying and shouting - so long as she is reassured with the returning (WIWO) and verbal then she is fine. Whilst we do need to respond to our children's cries (or shouts) for help/attention we do not need to be held hostage in a toddlers room for weeks or months on end.
As for waking baby, that's another issue. Some babies just sleep through noise once they are in deep sleep. But if this is a problem then perhaps a gradual withdrawal instead? So if Daddy already made it to the door today then continue with that and take small steps from there every couple of days.
It begins then with Daddy returning her to bed when she comes to him, verbal reassurance, tell her it is night time everyone is sleeping, she must sleep. Tuck her back in bed and walk directly to the door, he can say "I'll be right here, I'm with you, go to sleep" and park himself at the door. If he needs to lay down and doze I'd suggest doing it as close to the door as is possible, just inside the room.
After 2 nights he can move to outside the door. Continue the verbal reassurance from outside, if she cries and screams he can go in, reassure her and go back to his place outside the door.
If however he can't make it to the door due to the level of crying then he would need to start right by her bed where he has been up to now and take much smaller steps in the GW process. This is a longer process. A couple of nights near her bed, a couple of nights a little further away and so on.
There's some information here:
Toddlers: Walk In/Walk Out vs. The Gradual Withdrawal Method (HOW TO CHOOSE)What you (and Daddy) need to consider in choosing which method is the best for the whole family. At the moment everyone is losing sleep because your 2.5 yo is demanding Daddy be with her from 4am. You already know this isn't ideal. Being 2.5 is hard anyway, and can bring on bouts of separation anxiety, in addition she has been through some changes in her life which is a big deal. You need to be understanding and sympathetic to her need...but equally you need to think about everyone else's sleep. No one does their best parenting when they are exhausted from spending half the night in a toddlers room. Standing firm with WI/WO may be noisy and disruptive to all for fewer nights. GW may be calmer but take much longer.
I think I'd be tempted to give her one good EBT too. It sounds like she is tired and perhaps needs one catch up night with a little extra sleep. Maybe just 15-30 min earlier could do the trick to help her relax more and sleep better at that early waking.
Did you try the W2S by the way? This is probably the least disruptive of all the methods if/when it works. W2S was a life saver for me when we dropped to 1 nap, I couldn’t have lived without it. I didn't personally get success on the early waking with it, many do though so it is worth a go - and follow up with WIWO or GW if she still wakes. The beauty if W2S when it works is that she doesn't wake at all so you don't have to deal with the SA.