First off, let me confess that I don't know much about the Surface devices, but I do have a similar Acer laptop/tablet hybrid that's very similar in concept to the Surface. Mine also has two batteries, one in the keyboard and one in the tablet part; they actually both take the exact same battery, but I don't know if that's true for the Surface.
I'm going to take a wild guess here and assume you're running Windows (ha ha, of course you are). Windows will actually see two batteries when the parts are connected and will draw power from whichever one has the most (or any) charge.
So given a choice, it would be better to have the good battery in the tablet part, since you can run the tablet by itself, but you can't run the keyboard by itself. That being said, with Tom's statement that the keyboard battery is easier to replace, it will almost certainly do what you want if you just put a new battery in the keyboard; that is, you would be able to run your Surface from one room to another without a problem. The catch is if you do put it in the keyboard, you'll have to leave the tablet attached in order for that to work. If you put it in the tablet, then you can move it whether or not it's attached to the keyboard.
The short answer is, yes, you can replace only one of the batteries and it will work fine. It would be preferable to put it in the tablet part, but for your purposes and for ease of replacement, putting it in the keyboard should do what you want it to.