i really don't think that what you are suggesting will be that helpful.
the problem you are having is that she won't stay asleep if you put her down? so at 8pm, she falls asleep, but when you try to put her down in her cosleeper, she immediately wakes up unless you are right there with her? is that the problem?
your current solution is to have your DH wear her in a moby until when? when you all go to bed? and this is no longer working because your DH's back hurts.
if all that is correct, you have two different solutions:
one is to get your DH a carrier that can actually support her weight and is comfy for him. since he is already good with the moby, i'd suggest a woven wrap. they are meant to support much more weight and are more comfortable for the wearer. plus, you can try out some back carries, that allow your dd to feel close and secure, but give your DH a bit more freedom to do different things. a carrier like a mei tai, ergo or something similar might work too... the best thing would be to go to a store that sells different types, and try them on, or if possible, go to a babywearing meeting.
the second is to go to bed at 8pm with your baby. set yourself up comfortably with a couple of books, a cup of tea, a laptop with headphones, etc. i know how you feel about wanting some space before going to bed, but at 4 months, that just might not be in the cards for you. anyway, do whatever you do to get her to sleep, nursing (keep trying, babies' habits and desires do change over time), rocking, swaddling etc. then, get comfy. read or whatever... do this for a few nights, just enjoy the downtime. DH can join you and you can watch something on a laptop or whatever. now, here is where you have to remember that nothing stays the same forever. when she is deeply asleep, you can try getting up. at first this might only mean you get up for 15-30 min, until her sleep cycle is over. but some days you might get lucky and get an hour or two, or even more.
the most important thing is that you identify what things she needs to sleep best, and provide the best sleep environment you can. for my dd, she needed swaddling (still does!) and sucking to stay asleep. white noise, motion, etc weren't so important to her... so we were always able to get good stretches of sleep out of her in the evenings before my bedtime just by swaddling her well, nursing down and then giving a paci. this all does pass... it might seem daunting now, but it will pass.