Hi there. Hugs, I know it's really tough. DD was probably one of the worst sleepers EVER, but things started to get a lot better after 3 months when we figured out that she needed longer A times. If A times are too short your lo might be tired enough to take a short nap, but not a good long one...then as the day goes on they'll have too little daytime sleep and be overtired. It sucks.
If you stay committed to independent sleep (not rocking her all the way asleep) and find appropriate A times, my guess is things will get better soon. A lot of people find that 45 minute naps = undertired. You might try moving toward a 3.5 hour schedule if you haven't already.
My dd also likes some intervention to go down and it would be her preference to be rocked to sleep--STILL.
I set a limit of about 3 minutes rocking chair time until she goes into the crib, no matter what. At that point she'll usually talk to herself and grab her paci and go off to sleep. If she needs a bit more settling it's usually just a hand on the back. This is a baby who would scream for 20 minutes before every sleep at 2 months and wake up 20 minutes after being put down. If she can do it, your baby can definitely get there too.
