Hi canadawp
My best suggestions would be that you talk to your company to see the possibility to get and LMIA and that way possibly being candidate to apply for PR through EE.
If that’s not an option, yes your wife can apply for a study permit and if it is approved you will be able to apply for an open work permit. I just advise to do it ASAP, before your PGWP expires.
If you’re visa will expire it isn’t a problem as long as you don’t have to leave Canada and then coming back.