My DD is 4 months old now and she was the same - reflux/wind and startle reflex meant she never slept on her back - in the pram flat she screamed, in the car seat she screamed, if I put her down in moses basket she screamed ! From newborn she only wanted to sleep on me (preferred upright) or in a sling in the daytime and in the night time she had to be nursed to sleep, held upright for ages, then swaddled in a straight jacket style velcro swaddle and be next to me in the bed, on her side with rolled up blankets wedged holding her on her side - that is how she wanted to sleep! At the beginning in the day time she wouldn't even sleep on me if I was still, I had to be constantly bouncing on a pilates ball or rocking her/walking around etc.
I was tearing my hair out & so exhausted because I was recovering from a 40 hour labour and the pilates ball bouncing was hell for my stitches ! So my MIL suggested I put her on her front as that was how all her babies were put down 30+ years ago and she said she thought it would be relief for her tummy problems and solve the problem of her arms always waking her up. So I did and she slept SO much better...but then I had the problem of in the daytime like you said when I tried to put her down, as soon as she knew she was being put down in her Moses Basket or pram or anywhere (on her front) her little head would pop up and her legs would wiggle and she'd be doing tummy time AWAKE ! I tried everything like you said...waiting until she was super deep asleep etc...then sometimes she'd stay down for 10 mins then up her head would pop again and she'd be awake crying for me !
So I just held her...and put her in a sling and picked her up when she cried and hoped hoped hoped she'd grow out of it....I kept trying to put her down though every now and then to see if she would take to it...and it did start working slowly but surely...first she would sleep in Moses Basket and if she stirred we'd pat her bum quite firmly which made her sort of 'bounce' on the mattress and that soothed her back to sleep. She did these good sleeps in the Moses Basket for a while until she decided NOPE never doing that again and it was back to a few weeks of holding her for naps....But she slept so well at night (slept through at 6 or 7 weeks I think once she was in a dropside cot next to me) So I thought, well if she wants me to hold her in the day, that's what I'll do !
Then I realised that maybe she needed to be put down in the same place she sleeps at night for day time naps so I started putting her in her cot once she'd fallen asleep on me and patting her ...then I'd slow the pats down...then just have my hand on her bum...then leave her altogether and she'd stay asleep!
After a while of trying this (not always successful...if it didn't work one day I'd just pick her up and let her sleep on me) she started to 'accept' that she was going into the cot ...I would kneel on the bed next to the cot and slowly ease her down onto her front... sometimes she would open her eyes and realise she was going into her cot but she'd accept it and fall back asleep. Some days she'd scream bloody murder and would not let me put her down but I just held her and then tried again the next day!
Suddenly around 3 months she started resisting being nursed to sleep...so I had to only rock her...took ages...exhausting ! Then around 3.5 months she started resisting being rocked and cuddled to sleep too !! Bringing her knees up and bucking on my shoulder and crying when I took her into her room and started swaying with her on my shoulder. So I thought ok, she knows we're in her room & she knows I'm trying to get her to sleep and she's crying because she's tired..she won't respond to anything I usually do to get her to sleep, maybe she is ready to try and fall asleep on her own.
This was pretty scary for me as she used to be the sort of baby that as soon as she was put down ANYWHERE, ANYTIME she would scream as if she was dying, she was a proper velcro baby and just wanted to be on me and with me all the time ! I totally thought she'd just do that head popping up tummy time thing again and cry...never before had I put her in her cot awake but sleepy before...so I did a mini-version of what I do at bedtime (she has had a strong bedtime routine from 8 weeks) ..nappy change...draw curtains...dim lights...leg massage...take jumper or cardi off...put familiar music box on (Such a strong sleep cue for her now!) lay her down on bed and put her sleeping bag on...then cuddle and sway a bit to music box (now she doesn't resist she puts her head on my shoulder and starts falling asleep as she knows it's nap time) then as music box starts to slow down I put her in her cot on her back (I think this is KEY in getting a tummy sleeper to sleep in cot as she can see me & sees familiar surrounding of cot,bedroom etc rather than just seeing a big white mattress looming into her face haha!) then I gently turn her over onto her front, making sure her arms aren't caught under her tummy.
The first time I did this she wiped her face back and forth on the sheets, wiggled, fussed, cried a little bit and wiggled her arms around stroking them on the sheets...then I did shhhing and patting on her bum to soothe her but letting her fuss & cry a little bit cos I could she her eyes closing and she was trying to fall asleep so I didn't mind that was crying a bit, I could see she wasn't in distress. Then I just kept trying this at nap time...It didn't always work..especially not when she had a cold ! But after about a week or two...she now hardly cries at all (sometimes couple of mins, sometimes not at all) when I turn her over. She usually just goes right to sleep, even if she was awake when I initially lowered her into cot. It's taken 4 months but finally I can put my velcro baby down and she is learning to self soothe for day time naps...when I see she is tired (usually 1 hour ish after waking) then I start the short nap time routine..close curtains, dim lights, music box, sleeping bag, light off, cuddle, put down and it's been working really well !!
She is a catnapper though and has started to only have 30 or 45min naps which is a whole other issue in itself!!! But in terms of being able to put her down and her settle herself she's made amazing progress and I'm chuffed !! It just took patience, time, persistence and lots of reassurance and letting her fall asleep on me when she needed or wanted to...it didn't spoil her and I didn't create a rod for my own back. She just started self settling when she was ready and I kept giving her the chance and it started working.
Good luck and keep trying, it may take time but she won't be like that forever I don't think. Keep her close and hopefully it'll help her feel secure enough that when you do keep trying to put her down every now and again, soon enough she will feel comfortable to stay there !
Sorry about long message - typing quickly whilst baby is asleep !!