Yes you can request it, it's up to visa officer if they actually consider it or instead simply reject the conjugal app and tell you to re-apply as common-law.
https://www.canada.ca/content/dam/ircc/migration/ircc/english/resources/manuals/op/op02-eng.pdf
5.51. Switching categories between spouses, common-law partners and conjugal partners
Note that many people do very small court room/legal weddings only with limited or no family to get married on paper, and have no problems getting approved. It all depends on length and details of the relationship in general before getting married. Personally I can't see how a conjugal app can succeed when you do intend to get married someday, marriage is so easy to do right now, yet you are simply choosing not to for personal reasons. Of course ultimately the final decision is entirely up to the visa officer.
Would be interesting to see the exact wording of your procedural fairness letter. Usually in these situations the roadblock is not the relationship istelf, but the
legal barriers to becoming married or common-law. What you've described are difficulties that can possibly be overcome, not barriers. As was stated before,
many applicants stay in Canada for 1 year as a visitor and unable to work in order to become common-law.