I've gone through the Canadian border hundreds of times the only thing they ask you is sometimes is the license plate to make sure it's not a stolen car and you know your license plate for your vehicle other than that they never ask you for any information to show that it's your car, you should be good to go.
I have even driven up a brand new lease car from the u.s. to bring to Toronto Canada for someone to have for 2-3 years.
Don't offer any information they don't ask for just play normal and answer the questions they ask don't tell them oh it's a rental car.
And of course have a safe trip.
Just remember the passports or enhanced drivers license.
(Not sure if BC still works.)