Big setback with the Cockapoo

How old is Lucy now?  This may simply be that she's still too young to be 100% reliable. 
 
Usually when one of my dogs has an accident, I promptly take a newspaper, roll it up, and smash myself on the side of my head yelling, "Bad dog owner, bad dog owner," because once you have a dog trained or nearly trained, it's still very up to you to get the dog out.  The owner will get to a point where he/she knows when and how often each day the dog needs to go out.  Although Lucy has been using her doggie door, you still need to pay attention and notice if she hasn't gone through it for a while.  Also, where she has had some separation issues, she may not like going through that scary door to the outside without you.  Give her a break and take her out to pee.  As for two pees in the house in one day being classified as "out of control," with a puppy in your house you need to change your definition of outta control. 
 
 It's also good to look at health concerns--have you recently changed her food?  If this continues, you may have her urine checked for an infection. 
 
I'm concerned that you say that you "grabbed" her and that she whined and cried when you took her outside, and that you considered this to be because she is a wimp.  It doesn't matter what you think she is, what matters is what she's thinking, and that is that for some reason, you are scaring her with what you are doing, and you need to change the way to handle her.  She needs more gentle treatment and understanding, although you think you are treating her gently enough, she doesn't, and so you need to adjust. 
 
That you cannot look her in the eye without a pee taking place is not a great situation.  You want to be able to look into those beautiful little eyes and show her your love and feel her love for you come right back at you.  It may be beneficial to see a trainer in order to heal some of this. 

I've always found that after baths, my dogs are REALLY excited, and usually need to go right away.  Dogs, like children, will have accidents even after they're supposedly potty trained.  This doesn't mean you're a bad owner, or she's a bad dog.  She just needs some leniency. 

I dont really like doggy doors because I would rather know when my dogs have gone, or when they haven't, to spot potential health problems.  But if you have used the doggy door for 2 months and found it very useful, then it's up to you to keep using it.

My dogs still have accidents in the house.. once in a blue moon.. but they do.  So I just put them outside, pinch my nose, and get cleaning.

Nervous or wimpy dogs often pee/poop inappropriately after their baths.  It's stressful for them.  Unless you notice this behavior continuing, chalk it up to that.