My first reaction is that you're trying to see too much of Europe in the 3 weeks you are there. We just got back from spending 3 weeks just in Spain, and there are still areas that we didn't get a chance to hit that I'd love to visit. You'll be wasting several days just traveling by train/air instead of spending that extra time exploring one of the cities you are visiting. Cut your location list in half and plan to go back for round 2 later.
To answer your question though, it sounds like you already have plans to buy a full frame camera at some point. The 17-55 is an amazing crop sensor lens, and I love it for travel, but if you buy it now it won't do you any good when you eventually move to a full frame body. That being said, if you end up wanting the cheapest upgrade that will make the biggest improvement in your photos, get the 17-55 to pair with your T3i and be done with it.
If you decide to go full frame before the trip though, you have a lot of options. Before choosing full frame though, realize that you are adding a lot of weight to your kit by doing so. Full frame cameras are heavier, and full frame lenses are a lot heavier. The lightest full frame setup would be a 6D with either a 24-70 f4 or 16-35 f4 zoom, or a couple of shorter length primes (24, 35, 40, 50, 85 are all good options, but not as versatile as a zoom).
If I had to pick just one travel lens for a full frame camera (definitely not a crop), it would be the 24-105. It's just a very broad focal length, and while the lens isn't perfect, it does the job very well. Version 2 of the 24-105 is quite a bit heavier while not offering much better image quality, so if weight is important to you try to find a version 1. You can still get them new pretty cheap on canonpricewatch.com.
For Europe though, you will want to go ultra wide for anything indoors. On our trip I mentioned above, I took a 5D Mark II with a 24-105 and a 16-35 f4. 70% of my photos were taken with the 16-35, and only 30% with the 24-105. While there, I would carry the 24-105 if most of the time was going to be spent touring a city or countryside. But if we planned to visit a castle or cathedral, I would take the 16-35 instead because having that wider focal length is incredibly useful indoors.
A 70-200 wouldn't be a bad way to go either, especially for the Alps. Get the f4 IS model to save some weight. You'll be walking a lot, and carrying too much glass can make a trip miserable. If you end up with a 24-105, keep in mind that on your T3i that's a 38-168mm equivalent. So you can get most of the focal length of a 70-200 by bringing your T3i along and using the 24-105.