Hey there!
As far as sleeping through the night, not sure when that is going to happen. Babies sometimes need 1-2 night feedings until they are 1yr old. I do not think there is much you can do to speed that up any. She will most likely stop waking in the night to eat, when she is ready. She may give you some clues, like drinking less when she wakes for the night feeding. Not to say it never happens, but at this age most still need at least one feed in addition to the DF.
About bedtime. If she is on a 4 hour easy, going by your routine - I am guessing her catnap is around - 4p or 5p? I'll guess that it's for about 40 mins or so? So, she is then awake from 440 or 540 til 8pm, this is a really long A time before bed for 14 weeks. At 3months average night sleep is 10 hours but that is with 5 hours of day sleep.
On a 4 Hour easy, typically there are 2, 1.5-2hour naps and 1 catnap of 40mins approx. She isn't taking good naps - so she may not be getting even 4 hours of day sleep. So if you add the late bedtime - she may not be getting enough total sleep. If you feel its working for you, well then don't change it - but I would be curious to see if her naps would be better if she was getting more sleep at night. It may make a difference in the length of her naps, to get more sleep at night.
If you want indep sleep you may want to stop rocking, it may become a prop. You want her to get sleepy and fall asleep in the crib to associate that feeling with the crib not your arms, and not rocking - or she may grow to need them both to fall asleep.
Try experimenting with pat/shh and find a form of it that she finds soothing. Do your wind down minus the rocking, you can try sitting and singing or talking instead, put her in the crib with your phrase, then you can leave or duck out of sight. If she goes off to sleep fabulous, if she cries a proper cry, go to her roll her on her side and begin pat/shh. P/S her until she starts to settle and seems to be melting into the mattress, slow your patting and stop shhing, gradually slow the patting as she is settling more, when she seems to be ok, stop but keep your hand on her for a moment, then let go and watch. If she is going off to sleep, you can duck out of site or leave. If she wakes at 45 mins, you can use this method to help her settle again so she can go back to sleep.
Also, you may want to try wake to sleep again. I use method 2 - go in 10 mins before she usually wakes - so in your case try going in at 35 mins into the nap go in and gently stir her by gently touching her cheek, when you see her stir - then leave.
I think the key may be with the bedtime
I hope this helps some!
HUGS
Zoey