I would call the dealers' parts department and ask about cutting a key for your car. The TPMS error will not have an effect on the emergency key.

Before you do anything, compare the two emergency keys. If they are cut the same, then it may be that the door lock is messed up and needs a shot of graphite or other lock freeing liquid.

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Keeping it running is better than buying new

I also have an 08 XC70 - when I get a low tire pressure light the car does refuse to unlock with the remote key - fortunately my emergency key works fine.

My dealer refuses to recognize the locking problem, but it has happened to me three times. I had to have two tires removed and the offending wheels cleaned. The tires are good.

Your dealer would be able to get you a new emergency key by your VIN. If you do not have a regular dealer, you may have to prove your ownership with proper paperwork.

Perhaps someone can tell you if a locksmith can cut you a new key.

Someone may be able to tell you if that can be done.

Out of the box -

You could leave one of your rear windows open a bit and put in a lock button that you can hook with a coat hanger.

If you have to use a coat hanger on a closed door - my trick is to use two hangers - use the first is used to ge though he door seals, the second follows the first through the seals an unlocks the door. This has worked fo me several times on other people cars.

You can practice on a rear door to fine tune your technique before you need it. You need a place to hide two hangars under your car.