This is my story: the 1st licensed agent applied for VR before my wp expired and he said that I had to to wait for result of VR, I could not apply for a wp although LMIA letter approved. When my VR refused he said I still had 90 days to restore but I am out of status at the time my VR refused. I felt that he don't have much experience to deal with difficult cases so I found the 2nd licensed agent who has many experience with complicated cases. He said I could apply for wp before my wp expired with explanation letter and then we would submit a LMIA letter approval later. It's much better than applying for a VR
Yes, your case is almost the same. I think you should discuss with your lawyer again about applying for the work permit with LMIA receipt first in order to prevent out of status (if the VR refused). I am not sure at this step.
About processing time of LMIA, you can check at:
https://www.canada.ca/en/employment-social-development/services/foreign-workers/labour-market-impact-assessment-processing-times.html
It says that "from when it is received by Service Canada to when the decision letter is sent to you"
You can push your lawyer, ask them to call Service Canada weekly to check the status.
Hope it helpful.