Sleeping in Swing

Unfortunately not, baby is at risk of positional asphyxiation and it doesn’t matter how closely he’s watching, it’s silent when it happens. The fact that your SD is fine is down to sheer luck, a lot of babies weren’t fine and that’s why the guidance changed. Each to their own and the decision is yours as a parent but the guidance says no, this isn’t safe.

Can he trying a swaddle bag in the moses basket or crib? That can help them feel more secure

It is absolutely not safe to let a baby sleep in a rocker, carseat or pram (that isn't a bassinet). Swaddling bubba may work and is safer.

Please be really careful. Look up what happened to the babies in the rock n play device. Especially for a baby so young - it’s not safe. My partner has much older kids and told me that they slept all day in the car seat 🫠 nah, not happening here.

I’m gonna be the devil’s advocate here and say that sometimes as a parent, no matter the kids age, you have to make choices where your kids safety COULD be at risk.
And I think parents are way too fear mongered these days, and the societal pressure to obey and heed every POSSIBLE risk warning is ridiculous.
YES there are risks associated, aaaand in my opinion, I would still say there can be serious risks for when parents don’t get sleep and you should weigh the risks of both and decide what feels right for yourselves.
I had one of those breathing monitoring things for when my baby slept, and I let him sleep on his stomach.
I know everyone is gonna gasp and want to call cps, but I knew I would be alerted the moment he stopped breathing, he just would NOT fall asleep on his back, I NEEDED sleep, and billions of babies have survived sleeping on their stomachs, or cosleeping, over the thousands of years humanity has existed, so to me the risk was worth it.

We did the swing for a while for the same reason. Maybe a month then she didnt need it. Yes there are risks but there are risks in everything and caring for a baby without sleep also has serious risks. If the baby needs that incline more long term I might see about GERD issues with the doc and see if there are any special baby mattresses that provide more incline for GERD babies.

Also keep an eye out to support your husband even if it’s “his shift”, support each other to stay safe. Like I said exhaustion can cause accidents. So if it’s been several nights in a row baby can’t sleep ask if he needs to switch shifts for a night. My husband did this for me with my daughter during that first month and it helped. And we heard the same warnings and weighed the risks and even stayed up all night watching her for a while but it was okay for us short term.

Personally I'd look at the guidance about safe co sleeping and take that over sleeping in the swing that puts them in such an unsafe position for unobserved sleep.

You've mentioned in another comment about feeding/pumping but just for your information the guidance if ebf is to not pump until baby is 6 weeks as that's how long it takes to regulate supply and you can end up more likely to have mastitis or an oversupply if doing both.

Owlet socks aren’t a horrible thing, but can sometimes lull you into a false sense of security. Many times, they’ve alerted when nothing is wrong and cause more undue stress. This is a sucky time period for baby sleep. Give it time and lean on the help that is coming to visit.

I’m probably not the best person for this. But I co-sleep! And my baby is about to be 6 months old. Do what you feel is best. I only swaddled my baby in the hospital. My baby is currently laying right next to me in bed as I’m typing this. I just flop a titty over and she helps herself. Guess who doesn’t suffer from sleep deprivation? Me! There’s risks to everything! I remember my coworker telling me one of her kids only wanted to sleep in their swing! Hope you find what works for both you and baby

There’s pros and cons to co sleep as well. I personally could never do it because I was afraid of smothering my kids. I slept too soundly, even with babies. I also didn’t breastfeed. I made my bed be space I didn’t want to have to share with my kids, then I never had to break them of it. Each kid slept in our room in a bassinet until about 2mos old. They both “mostly” slept through the night by then and we used a monitor when they started sleeping in their own room.