What to use to tame my chicken tikka masala?
You're thinking along the right lines with using yoghurt. In fact because yoghurt is often used to tone down spice, you might find that you can use more than half the sauce, if you increase the yoghurt.
But you probably wouldn't want to double the yoghurt to make up for the missing sauce. Instead replace the sauce you didn't use with yoghurt. So if the packet contains 200 ml  of sauce, and you only use half of it, you'll be short 100 ml, so use an extra 100ml yoghurt. You might find that's a little runny, so might want to use a bit less.
If you have tomato puree/paste or even passata, you could mix some of that into the extra yoghurt. Most tikka masala recipes include some tomato, so this will maintain the flavour without the heat. Puree will have a slight thickening effect compared to yoghurt, while passata will be thinner,unless you simmer it long enough to reduce.