@Frazzledandtired91
What is your feeding schedule over the daytime as well as night? Is baby finishing the bottle fully or is some left in the bottle after a feed?
She drinks about 900ml by bedtime (at 9pm) . She's awake for 2hrs 15 and then she gets a bottle, sleeps for 30-40min nap in the day, and then repeat - so she's basically being fed every 3 hours
How do you get baby to sleep at bedtime and during these night wakes?
Feeding, rocking, singing, blowing on her eyes, dummy to calm her as well as a swaddle
What's baby's daytime sleep schedule? Where and how does she get to sleep?
She sleeps in her cot ^ see above for details do daytime schedule
2h15 awake window is very long for the age. Generally your awake windows want to be around double nap length, so 30-40 minute naps want 60-80 minute awake time. Over tiredness ends up causing restless sleep in the night with frequent night wakes.
I'd therefore shirt your 3 hourly repeating cycles down to about 2 hours. This will also help up babys daytime calorie intake. Four months sees a significant acceleration in baby's calorific need, it's not unusual for the to go up by 25%. Because baby's stomach volume limits how much milk can be taken per feed, it often needs you to feed more frequently to get more into baby.
Feeding, rocking, singing, blowing on her eyes, dummy to calm her as well as a swaddle
I'm not sure which it is? If you want baby sleeping independently (ie in the cot, as you mention) then baby needs to go from awake to asleep in the cot. So that means not feeding or rocking to sleep. Putting baby down already asleep is very poor sleep hygiene and it is part of the problem at this age.
That's not to say feeding and rocking to sleep are inherently "bad", they aren't. But if baby is going to sleep in your arms, it is better sleep hygiene to have baby staying there for the whole nap. This lends itself more to attachment parenting and not independent sleep.
If you want baby sleeping independently, do not underestimate how much work you have to put in to help baby sleep like this. It is very labour intensive and the easy option that requires less work from you is to attachment parent.
You will probably have better luck at this age to move daytime naps into something that moves. This allows for movement to sleep, but in an independent way. It also allows for movement to resettle and try to lengthen naps.
This option isn't available at night though. If you haven't already, then removing one side off your cot (to make a sidecar cot) makes night resettles easier to deal with. Dont try to put baby down already asleep, your focus needs to be on getting her to go to sleep in the cot. Swaddling will help reduce the startle reflex and encourage deeper sleep - use it consistently though (for the sake of Safe Sleep guidelines).
Also ensure dummy is being actively sucked until in a deep sleep, not sitting passively in babys mouth. With the sidecar cot you can cuddle into the cot to help baby settle in there, then extract yourself once asleep.
I'm not sure if you're feeding overnight? While it's good practice to always try to resettle baby back to sleep without feeding and to not feed at every wake up, it's quite reasonable to expect night feeds at this age so if you are not feeding overnight this may be the problem.