What you are saying doesn't make sense. Why making the QSW application process longer will guruntee you settle in Quebec? That's actually one of the reasons a lot of people are leaving Quebec.
I spoke with a CIC agent and she said they don't even know if I have a CSQ or not as that's the Quebec's govermengo thing. As long as I have not applied for QSW in federal it doesn't necasirly show any intention that I am going to settle in Quebec. It is possible that you had decided to settle in Quebec but for any reason your situation has changed and now you want to live somewhere else in Canada...
You are getting misinformed in this thread by someone who do not know the difference between CSQ and QSW and quirks of Quebec immigration.
I understand your situation exactly. Getting a PR thru QSW is a very long one and nowadays it has been getting longer than ever before. High skilled talented people should never opt to work in Quebec. They should leave Quebec immediately after getting 1 year Canadian work experience or if and when they get enough points for Express Entry.
Having CSQ does not deter you from applying for express entry. CSQ is a not an immigrant application. QSW is an immigrant application. Now in EE there is a place where it will ask if you have a CSQ. You should tell yes and enter your CSQ number there. You should also add a letter of explanation citing all the reasons that you do not want to stay in Quebec. This is the process when you apply for EE from Quebec.
Now getting back to your individual situation.
Plan 1
Since you have got your CSQ, you employer do not need to apply LMIA or CAQ to extend your permit. Your employer should apply in Canada employer portal ( pay 200$ fee) and get a job reference number for you. In parallel you apply for QSW and following QSW immediately apply for work permit extension yourself with the employer reference number and your QSW AOR letter.
Now the problem here is since QSW is a paper process, you will have to wait longer to get AOR for QSW. If you have time, then go thru this process. Once you get your work permit extended. Say goodbye to Quebec and your a-hole employer, move out, apply for EE and live happily ever after.
Plan 2
If Plan 1 is impossible, your only way to stay in Canada is to move out of Quebec, apply Express Entry asap and once you get your AOR apply for BOWP. Because staying in Quebec and applying for BOWP will most likely get rejected.
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If your employer had helped in your closed work permit process, you could have stayed in Quebec and applied Express Entry, but since your employer is not helping, the only way is it move out of Quebec and apply EE asap and get BOWP. Change to visitor visa if possible to help in this transition.