We are not able to settle our 7 week old to nap during the day. His best day he slept 4 hours in the day, his worse none at all, not even 5 minutes. As a result he is constantly overtired & only stops crying to feed. This has been going on for 3 weeks. He does sleep well at night but not in the bassinet and not on his back. The only time he will sleep is on his side in our bed. My question is what is worse for his health; zero to fours hours of sleep every 24 hours or sleeping on his side in our bed?

It is difficult to provide a direct answer to your question.
I’m sure it is distressing for you all having difficult getting your baby to rest & sleep.

Have you been in touch with your Maternal Child Family Health Nurse in relation to these sleeping/settling issues?
There are many services across Australia that can provide information & support.
If you are concerned that there is a health reason for your baby’s sleep issues please see your GP or Paediatrician.

There are some risk for babies when sleeping in an adult bed.
While Red Nose recommends that the safest place for your baby is in there own sleep space we do have some information about how to reduce the risk to your baby when sharing a sleep surface.
https://rednose.org.au/article/is-it-safe-to-sleep-with-my-baby
https://rednose.org.au/article/Co-sleeping_with_your_baby

While sharing a sleep surface is practiced by many families without any adverse outcomes, there can be specific risks, especially with a very young baby less than 3 months of age.
It s also recommended that baby is still placed on there back for sleep, with bedding modified so that baby’s head & face is not covered & baby is not wrapped/swaddled.

Please have a look at the information provided to make a decision about co-sleeping.
https://rednose.org.au/downloads/Co_sleeping_for_families.pdf