Whilst I feel immensely fortunate for having an independent sleeper by that age, as I said before we were not without our tricky phases. Mine would not feed to sleep, nor could he be APOPed, wouldn't fall asleep in arms, in the pushchair or sling, refused a paci and at 3.5 months ran into his 4 month regression early, we had 2 months of short naps, 40 min each, 5 times per day.
If you are getting short naps you will need to put in an extra nap or two or you'll be unlikely to get through to BT.
I want to suggest doing WD in the sling, not for her to end up having an additional sling nap but because this seems to be the place she is calmest. Do you hold her through the sling so she can feel your arms? This might be a step to try, holding so she becomes accustomed to that feeling (with a view to feeling calmer in arms when not in the sling) and using either shush or a key phrase to repeat. I would also 'hold' her through the sling and use shush or a key phrase on the other sling naps in the day during her wind down and nodding off period.
Honestly I am surprised you have not seen any improvement yet and am just trying to offer suggestions to help her associate something soothing with sleep time.
Does the sling smell? Does she rub her nose in it? Just wondering if she is associating the smell of the sling with sleep and has become attached to it? There could be a way of testing perhaps by wrapping her somehow in the sling but not wearing her, doing the WD, putting her down in the cot with the sling - DO NOT LEAVE HER - it is not safe to leave her this way you need to stay throughout, but you could observe to see if she was at all calmer, if she fell to sleep you might be able to remove her from the sling wrap to leave her safely in the cot without additional items in with her - although chances are she'd wake. I wonder too if you can return and put the sling near her face/nose during your W2S attempt, see what happens. If it turns out she is calmed by the sling smell or fabric we can look at introducing a lovey for WD, she wouldn't be able to be left with a lovey yet but mine fell asleep with a lovey very young, I just removed it after he nodded off.
Just a thought.