Hi
Your employer can apply for a labor market opinion(LMO) for you. Since you were hired on a post-graduate work permit, they do not have to show recruitment efforts which is good for you. The turnaround times for LMOs is around 14 days(could be longer depending on where you reside) and if the application is done right, you should get a positive LMO and then apply to extend your work permit.
1. The OP already said that the employer won't support a LMO.
2. Secondly what province can you get a LMO in 14 days, most provinces it is between 10-12 weeks for an approved LMO.
3. The OP is working Part time, HRSDC will not issue LMOs for part time work.