I've started giving her baby rice in the evenings and she has 5 bottles a day of between 180-200ml each
So you do recognise that this will, in part, be due to increased hunger. You are right, it often is around this age. I wonder why you choose to offer more difficult calories rather than just giving more milk?
Gram for gram, the most easy-to-access calories for baby is milk - formula or breast. Calorific need often goes up around the 4 month mark, and quite significantly too. How much milk baby can take in one go is limited to the size of the stomach.
The way to get more milk into baby is to offer milk feeds more frequently, so that there are more feeds per day. If these extra calories are not all given during the daytime, you will get waking in the night.
Is baby draining a 200ml bottle? That's a very large amount in one go at this age. My 91st centile chunk was given 180ml bottles at this age and usually drank between 120-150ml. But, she had 7 or 8 bottles per day. I used to feed 2 hourly through the day - 7am 9am 11am 1pm 3pm 5pm 7pm and then sometimes an 11pm feed or night feed.
Baby being hungry is not seeking to 'feel full' - it is calories baby needs. So if baby in-naturally has solids before the digestive system is ready, those few spoonfuls provide a few calories. But an extra bottle full of milk would have given baby far more calories. So there is no benefit in baby "eating" this young. Indeed, early weaning (as in 6-9 months old) often does result in unsettled nights precisely for this reason - when the calories from milk feeds reduces.
Maintain those milk feeds, increase them, They are necessary.