In theory that should be fine, but there are definitely risks:
- If something wonky happens at the tire place and the car gets damaged there, there could very well be a mess
- If you get into an accident with the car and it has tires on it that weren't installed with approval from the rental agency, it could very well void any extra insurance you purchase from the rental agency (if applicable)
Why not just ask the rental company what to do? If you're picking up the car anywhere near where it snows, this won't be an abnormal question for them.