Not sure about the extra charge but the reason hospitals make you room with baby is because the first few days of bonding is really essential so you can do as much skin on skin, latching, snuggles as possible as these will promote baby's growth. Nurses can help you with bathing, diaper change and feeding. Baby will mostly sleep so you can still rest while hugging baby.
I've not heard of such extra charges tho, my SIL was at NUH, didnt get charged for tt. I was in private hospital, nurses will only give baby when it's time to breastfeed which was every 2 hrs so rather than sending to nursery at night and constantly get woken up, I slept with baby on bed.
You can ask the nurse to bring baby to nursery so you can rest/sleep. But they will bring baby back to you during feeding times. Can always press the bell for assistance if you need help with baby. They are quite helpful during my stay.
never heard of extra charges (but I wasn't with govt hospital), but they are always ok to have baby back in nursery if we need to rest