Salam
sorry I didn't check this forum over the weekend.
I heard from non-Persians who tried that it does not have any impact on future visa. It seems that it is a very common thing and they know every year a lot of people from Canada go to the border for Flag poling. Some friends of mine actually tried and could get visa easily afterwards, but none of them were Persians. For Persians, we are going to have issues getting visa, no matter if we do this flag poling or not!
For the process, yes you will receive a letter showing that you were denied to enter. But I don't know what they write on that letter, sorry. But, I think every time you have to mention you were denied to enter to the US because of flag poling. It is going to be in your records, but again it seems it does not have any impact for future visas. Every one says it is harmless. I don't know.
Some other options
1- I don't know what does "get incorporated" mean? Is it something work related? I know that there is an urgent processing of PR card for those who are in rush. See this link for the conditions:
https://www.canada.ca/en/immigration-refugees-citizenship/services/new-immigrants/pr-card/apply-renew-replace/urgent.html
Email them and explain it is urgent for you. They may give you a time quickly. If not, you can think about other options.
2- I knew that in some cities, they used to offer in-land landing on a walk-in base only for some specific days in a month. For example, Ottawa office used to offer landing on a walk-in base twice per month. However, I heard from another person they canceled it recently. Raise a CSE and ask them if there is still any office that offers walk-ins or not? (Even though, I think plane ticket to mexico may be cheaper and faster for you that going to Ottawa from Vancouver!)
3- For travelling to Mexico, are you sure you don't need a visa? I know that those who have PR don't need a visa, but not sure you.
Sorry, after typing this, I searched for you and found this link. Thought sharing with you instead of removing the comment. It seems you don't have any issue entering the mexico.
https://consulmex.sre.gob.mx/toronto/index.php/en/servicesforeigners/doclegalization/52-conservices/225-visitors-who-do-not-require-a-visa-with-a-stay-up-to-180-days
4- If you decided to go to Mexico, you can look at some cheaper airlines as well. For example, check Air Transat. It is a small Canadian airline. I have never used them myself. So, I don't know about the quality, but I heard that they have cheaper tickets than westjet and aircanada.