Now, do you think I shall go ahead and file for PR before I receive my extension approval/ or shall I wait to get the extension approval and reject this invite and later try to obtain another invite ??
No problem to proceed with the eAPR.
My worry is - I was reading about this Dual intent - which is legitimate in principle but often doesn't work and IRCC tends to reject temporary visa applications, which in my case would be my extension rejection, so that way I will be out of status, given the fact that I'm already on implied status.
Dual intent is only a problem if you are coming from outside Canada and the VO believes you will not leave Canada when your work permit expires.
If your TPW extention application is rejected as incomplete, you would be out of status retroactively (doesn't happen too much). But it is refused as ineligible, your implied status only ends when you are notified of the refusal.
Please advise what do you think, is there a chance that my work permit extension will be impacted if I file for PR (I already have an invite, so I guess they anyway know that I want to apply for PR).
No problem. Implied status is legal temporary status, that is all that is required if you are living in Canada.
As for the suggestion to apply for the BOWP - since you do not have a valid work permit at the moment, you are not eligible.