I mentally walked through scenarios that would lead us into our emergency fund, and tried to figure out how bad things would need to get for us to use our funds to make these decisions.
I concluded that we need to be conservative with the first three month's funds, but we can really relax for months 4 to 6. We will only hit those funds if we have a major disastor that isn't covered by unemployment insurance, short-term and long-term disability insurance, maternal leave, life insurance, or health insurance - or if I lose my job AND we have a major emergency at the same time. We also know that we can sell our house and buy something smaller, my DH can go back to work (he is currently a SAHD), and DH and I have both been poor before - so even if we start to run out of money, we know how to handle the situation. Basically, the 4 to 6 month cushion isn't intended to maintain our standard of living - it is there to help us transition to a lower-cost lifestyle after a major event (our house burned down, I was fired and then hit by a car, I become pregnant with quintuplets, the country hit a Great Depression, you get the idea).
Our choice is that we will put a conservative 3 months into a high-interest savings account (we are just starting our emergency cushion - we just got back on our feet from the mild poverty mentioned earlier). So we would probably include the high gas costs, etc., and be really conservative in our estimates here. In reality, I just round our expenses up to the nearest thousand and call it good . . . we're pretty lazy with our budgeting, and it's all really a trade-off; if we save more, we get more security, but if we invest more we get more money. I don't have strong feelings about which is more important, so I don't worry about the details (you may want to consider if you really care, and if you don't, just do the same).
We will then put another three months into some other investment vehicle - we're still figuring out what, but we're going for something more volatile with a good chance of higher returns over the next ten years. If I lose my job and my cushy benefits, we'll probably wait until the market seems healthy (sell high) and then move the money into a more conservative account to protect our longer-term cushion until we feel as financially secure again.
Maybe it would help to think of the extra security you get from saving a little more as something you are purchasing, and weigh that security against other things you want (your renovation) and decide which you want to pay for first. You can always save a little more later, if you still want a larger cushion once the renovation is done. On the other hand, renovating has opportunities for all kinds of little surprise expenses to crop up - so maybe you want to pay for the extra security in your budget first. It really is a personal decision. Note that the car breaking down is just the kind of emergency your cushion is designed to protect against - most people wouldn't plan this far in depth unless they had a car repair buget item (we do, since we bought a junker car - repairs are inevitable so we save for them each month) or unless they knew to expect a specific repair in the next year or so.