My firstborn was like that and cried a lot in the bassinet. He actually had reflux and laying him on his belly helped a lot, however that is not really easy in winter and the bassinet. Still, try, and try a pacifier, too, as these two things helped us.
For a travel system, I would go with a forward facing model, as that also helped my son to be happier in the stroller - he was fascinated by the world around. Definitely do not use the car seat instead of the bassinet, it is not healthy. Check strollers like the Oyster Zero or the Valco Snap line as they are light but roomy enough.
From the reversible options, I would look at Joie Versatrax / Joie Finiti as those sit very upright and are not crazy expensive.
A Joie car seat fits on Joie / Maxi-Cosi / Cybex style attachments, so it means all strollers mentioned accept Joie car seats, and they are the least expensive while still good and safe. You will need adapters, though.
One more thing, definitely check second hand strollers, too. I did so with my firstborn, as I did not know if he'll want to be in the stroller even. If your daughter will not, at least you did not invest in a new stroller but saved a lot of money as well as wes bein eco. ^_^ Also, don't be afraid of "rattly" strollers, even a simple umbrella bugy could work here - my oldest loved the rattling and if he fell asleep in the stroller, then on bumps and while the stroller was rattling :-D