If, however, the new one comes through and I do switch, do you know what the process would be to apply for a new LMO with the new company? How would I address the issue that there will be 2 applications for me, through 2 companies, one after the other?
I don't think there is a problem having 2 employers apply for your LMO but I think that not showing recruiting efforts to get the LMO will only work for the employer you are currently working for. You can ask Service Canada about that as well.