How do you keep rocking her in the crib?
Mine was swaddled and when I put him down in the cot (I had a travel cot in the family room in those early weeks and a hammock, where he was not swaddled, for night sleep but I also rocked in the cot at 9 months+ when he was transferred to a proper cot-bed) so I put a hand on the 'bundle' like the nappy area and used a firm hand and just moved it back and forth a bit, it is very small rocky movement, at times almost a vibration rather than a rock, bit like how I imagine it feels in a car seat. it doesn't take much hand movement to produce movement in a little bundle. Mine slept on his back (he would have chosen to tummy sleep but my fear of SIDS kept him on his back) in those early months but later, when he was much bigger, in a cot and could turn to the side himself I put my hand on his hip and rocked forward back there.
That sounds like I rocked for months on end, I didn't, but of course they all have phases of needing extra help (illness, teething, developmental leaps, SA...) so it was this type of rocking I returned to rather than patting. You can produce a rock motion by rubbing with your hand but usually my hand would be firm.
Our wind down always included rocking, with a song (so kind of dancing) and then I'd transfer him to the cot and either help with rocking if needed or leave him.
I didn't have a wedge to raise the cot until mine was older but our hammock gives a raised sleeping angle which helps with reflux, have you considered using a wedge in the cot?. He found it more comfortable to sleep there. It also provides a gentle movement if baby moves or kicks their legs or startles/jolts, perhaps not at 3-4 wks but when they are a bit heavier so that was useful too. I put him in without rocking when I could, so the hammock was still rather than rocking, this helps to reduce and wean.
One thing I know I did wrong was I did the 5 S with him on the wrong side, it is supposed to be the LO's left side which keeps reflux down and lets gas out, I didn't realise that and held him swaddled on the right side instead...although he went down on his back when I was teaching him to lay down and sleep.
I'm really not sure I could have been so successful if I'd had another child to care for at the same time, I was able to put my entire attention into DS and gentle sleep training, it did take lots of time and effort but ultimately it 'gave back' that time because he was an independent sleeper early on.