My first instinc is to say that she is over tired. Starting the day at 5 is too ealy, once fed she should be back in bed until 7. Don't keep her up for activities early in the morning your setting up for trouble. Does she eat at 5? If not you could try the wake to sleep technique. YOu go in a half hour to an hour before she normally wakes up, reswaddle, give the paci back, anything to make her stir just a little and then leave her to fall asleep again. What it does is make her restart a new sleep cycle.
My son did start waking early around 3 months too, we did the wake to sleep, which worked for a period of two weeks. Then I just started waiting longer to go in as he just fussed. Sometimes he'd fall asleep others he'd start to cry.
YOu mentionned that the shush pat doesn't work on your LO, Tracy did mentionned in her third book that Touchy babies may find the pat too stimulating/distracting to fall asleep. Try just rubbing her back or belly; or simply keep a hand on her back while shushing pass her ear.
This usually totally upsets her and she starts crying....then screaming. Even after 20 min (I have not been able to wait the 45 min, she starts to gag and cough after 20 min) she is still screaming and shows no sign of getting tired. I will pick her up and pat her until she is calm but once I lay her down she starts screaming again...do I pick her up again right away or do I have to wait?
You should try to calm her in her bed before picking her up. At the age she is at you should keep her in your arms until she is calmer and just snuffling the cry. Then put her down but stay with her.
Hope this helps, I gotta go my son is waking.