Can my 12 week old sleep in a separate room for day naps if we use a monitor and sock monitor? She does have reflux but I monitor her regularly. It is so hard creating an environment conducive to sleep in a shared space with a toddler. It is affecting everyone.

Red Nose promotes following the Safe Sleep recommendations for all sleeps.
https://rednose.org.au/article/red-nose-safe-sleeping-recommendations

Many parents are unable to room share for all sleeps..
Some parents find a portable cot (that meets Mandatory Standards) is helpful to continue room sharing till 6 mths.
Ensure doors are open & you are able to hear her, & when you wake up during the night, check on her.
No parent is expected to sit up all night/day watching baby.

Important to maintain all of the rest of the safe sleep recommendations, in particular:
  always place baby on back,
  keep head & face uncovered (nothing loose or soft in the cot)
  provide the safe environment (cot with firm, flat(not elevated) mattress.
Red Nose has the following information

“Parents are not expected to observe their baby constantly. If your baby is sleeping in a separate room check baby regularly to ensure that the baby remains on the back and the head and face remain uncovered (as baby grows beyond 5-6 months they will move around the cot and roll over; settle baby to sleep on their back but let them find the sleep position they feel most comfortable in. A safe cot and safe sleep environment is still necessary for babies older than 5-6 months).”
https://rednose.org.au/article/room-sharing-with-baby