Yes, and this is one of the main results in a paper I hope to put on arxiv very soon. I wrote about this result in a previous mathoverflow answer [here][1]. You are right that the way to do it is to focus on maps between cofibrant objects. This result was also known to Clark Barwick, and probably to Cisinski, as I discuss in my other answer. That's one of the reasons it took me so long to write it up. I didn't know why anyone would care. But, in the paper I'm finishing, there are tons of applications of this result. If you want to correspond more, I would be happy to.


  [1]: https://mathoverflow.net/questions/300879/left-bousfield-localization-without-properness-what-is-known