Baking – How long can I store homemade caramel or butterscotch sauce

Not really. The sauce must be refrigerated but the butter will solidify when it cools. When you warm it up the sauce will break.

What I do is warm it very very gently in the microwave until it is liquid. Then reemulsify it into a single egg yolk. It costs an egg yolk but the sauce is good as new.

If you make caramel acidic, it won't harden. So use cream of tartar (neutral taste) or lemon juice (easier availability) to create a non-hardening sticky caramel fluid.

I am not sure if you can add dairy (milk, cream, butter, etc). to acidified caramel, but there is a small chance that it will curdle, so I'd advice you to use clear caramel (browned sugar and water only).

I don't know how much acid you need to keep caramel at a certain viscosity at a given temperature, you'll have to experiment for that. Or maybe somebody else can supply this information. IIRC, a good starting point is a tablespoon of lemon juice per 100g sugar, but this is very imprecise.

The other option would be to make a semifluid sauce (cream, or starch-thickened milk, or diluted syrup) and add caramel to it, but the taste would be much weaker. My preference would be the clear caramel + acid route.