Both of my grandchildren slept for several months in their car seat. My granddaughter was eventually diagnosed with asthma and I think that she could breathe better in her car seat. My grandson was eventually diagnosed with GERD and I know he spit up less sleeping in his car seat.
We just put the car seat in the crib and let them sleep. I suggest that if she sleeps this way then let her sleep in her car seat.
I also suggest that you could let her sleep on her side or tummy. Have you tried using tightly rolled up blankets on either side of her so that she stays on her side? You can also buy foam wedges to help.
For years and years babies were allowed to sleep on their tummys. Yes, a very few babies died of SIDS. Less than 1% I think. I suggest that tummy sleeping done appropriately is OK when that's the only way a baby will sleep and especially after they're able to roll over.
They have to be layed down on a firm mattress with no pillow or soft blankets/crib fenders. Their neck should be strong enough to lift their head and turn it to the other side.
I've attended a few crib deaths and all but one had a crib full of soft things. The one that didn't wasn't identified as SIDS. Cause was listed as unknown.
I also suggest that when you wait until she's tired that she has no patience to comfort herself to get to sleep. Start a routine that includes regular sleep times before she's so tired.