I don't recall anyone asking that one before!
Tricky.
The slight difference in the overall diameter of the space-saver is dealt with by the car's electronics up to a maximum speed of 50mph/80kph. Thereafter it's not and that's why you can get transmission wind-up which may burn out clutches in the autobox unless the car is running on grass, ice, gravel, sand - any surface where the wheels can slip and self adjust.
Whether the system will accommodate a difference approaching two inches, I simply don't know.
The standard 235/65/17 29 inch OD will do 695 revolutions per mile versus your new size tyres of 31.6 inches doing 636. That's nigh on 10% difference.
https://tiresize.com/comparison/
I use and recommend tyre strings. Quite often you can fix a puncture without even taking the wheel off!
I've even used four or five strings to fill quite a big hole before now. Needless to say that tyre was a write-off but I did several hundred miles on it at sensible speeds until my new set arrived.
I like to use a little rubber glue with the string. Some kits have it, some don't.
I haven't used these but they're worth looking at:
