It is true that you have a greater number of gears with a triple, but a compact trims weight, and with the substitution of a 30t rear cassette (which I would ask the LBS to do at purchase) you will have as low a gear as a 30/26, pretty near. I bought my wife a Jamis Ventura (aluminum) with that cassette and she's been able to get up the same hills I can with a 34/26, just slower. Total weight is 23 lbs., which is not super light, but it was an inexpensive bike and since she's just riding for fitness and enjoyment she doesn't feel that it's heavy, just puts it in the 30t and gets there.
The overall weight of the bike makes a difference in climbing, and the weight of the wheels makes an even bigger difference, but that only matters if you're trying to go fast. If you wish to lose weight by cycling, the first priority should be comfort, and a close second should be gearing. It is the amount of time that you spend at a relatively low heart rate that will allow you to spend more time on the bike, so you have to feel comfy and not afraid of hills, and not be stuck with a gear that makes you struggle. So if you try a compact with 34/30 low gear and it doesn't feel like you can handle it, a triple might make more sense. Or look at cyclocross bikes which typically come with components that allow lower gears off the bat.
I have carbon and steel road bikes and an aluminum mountain bike. Which you prefer is really up to you. Carbon frames in your price range are likely to be mid-modulus rather than high-modulus and a bit heavier than the really light race frames, so if your dollar gets you into an aluminum frame with a lighter group and lower overall weight AND you feel comfortable, you might be happier. Once you've gotten a bike that feels good and keeps you out on the road, you'll learn much more about what you wish you would have done and you can start looking into the future at your dream bike.