We removed the bib for sleeping in bed as well, but I had some spew cloths, bits of muslin/flanelette stiched together with a rolled edge overlocked, and just put them ontop of her bed sheets, more often then not it would catch the chuckies in bed and the sheets didn't need changing as often.

We wrapped for quite a while and had a huge stash of wraps so if she got the wraps it was a change before the next sleep.

Good Luck I hope you find a solution that works for you.
xxoo