less than 3 hours of sleep per day,
Are you saying your baby is only sleeping for 3 hours in 24?
Assuming baby is sleeping for more than that over 24h, you could lie down when baby goes to sleep. You could even cosleep for naps.
If you feel baby isn't sleeping well, try swaddling baby in a giant muslin or cot sheet - the security helps baby sooth. Also try giving baby a dummy. And feed, feed, feed.
Baby should be spending very little time in 24h being awake at this age. Whenever baby wakes it should just be a case of feed, wind, check nappy, have a cuddle, back to sleep. About 30 minutes awake, 1h at the absolute most. If baby is spending protracted amounts of time awake, that's a problem. Focus on getting baby sleeping more.
If baby is sleeping well but you are not, prenatal insomnia and not taking the opportunity to sleep when able to are signs of Post Natalie Depression. Your doctor would be interested in hearing about feeling like this, so book an appointment. (Catastrophsising is also sign of PND)