This is the criteria for BOWP according to IRCC:
"To be eligible, you must:
- live in Canada (Quebec not included)
- have a valid work permit that expires in 4 months or less
- be the principal applicant on your permanent residence application
- have submitted a complete application for permanent residence and passed the completeness check
- have your acknowledgement of receipt letter
- we send this letter to your account after we get your permanent residence application."
What makes your think you are not eligible?
if you are in Canada, have a valid work permit and you are the principal applicant of your PR application and have received the AOR, you are eligible.
Where did you get this from?:
"I was hoping to apply for a BOWP but it seems impossible since my current work permit expires in less than a month." --> FALSE