Is is possible to apply for a temporary work permit while being sponsored? Or would that be redundant because the processing times would be about the same?
You could but you need to find an employer first. If your occupation falls under NAFTA, it would be enough for you to have a letter with a job offer to get a work permit. If not, the employer needs to advertise the job and prove that no Canadians or PR's are interested or qualified and then he can apply for a labour market opinion saying that he already advertised and he is paying market wage and they will decide if they will allow him to hire a foreigner. If he gets the labour market opinion, you can use it to apply for a work permit.