Well there are several options. You could post a picture of your current CV/TieRod/LCA angles as well as clearance to the upper droop stop to get an idea of how far you are cranked already, and if you have any left to go. (Don't kid yourself and think the "leveling keys" you installed are anything more than cranked torsion bars, the result is exactly the same, and works by the same principle as cranking the keys. Probably 80% of the time the stock keys had more than enough adjustment to get you to the limits of the stock suspension without the keys, the only exceptions being higher mileage trucks with sagging bars, or trucks with heavy winch bumpers/plows.)
The first option if you are not too far cranked already, is to add a set of Cognito UCAs. They are ~$500. They will correct your upper ball joint angle, allow for easier alignment, and give you a little bit more downtravel along with a urethane bump stop instead of the metal to metal contact of the stock droop stop. This will get rid of the harsh slam of the suspension when it droops out coming off of bumps, such as over rail road tracks, speed bumps etc. They will allow you to crank your existing setup slightly more. Although this keeps the same disadvantages as your current setup: steep LCA angle effectively increasing the spring rate do to the shortened moment arm of the torsion bar, steep tie rod and CV angles which results in a weaker rating for these components (they are at their strongest when near level), and poor lower ball joint angle, resulting in increased wear. If you do not already, you will also want new, slightly longer shocks (or shock extenders as some kits include).
The next option is a 1.5" body lift. Personally I hate body lifts on 2500HDs and 3500s, due to the fact that the frame already hangs so low below the body of these trucks (2" more than a 1500, that is where the taller ride height of a 2500HD comes from), but if you must, the 1.5" doesn't show nearly as much frame as a 3". Although a set of step bars will hide the frame decently well. Just make sure to raise the step bars 1.5" with the truck so they don't hang down low, simple with most standard step bars, also a set of gap guards (or make your own from rubber) for the gap between the bed and frame so you can't see daylight through your truck. ZoneOffroad sells a 1.5" body lift for a 2003 1500, not a 2500HD. The 1500 kit will fit a gas 6.0 2500HD with little to no modification. The 8.1 and 6.6 take a little modification to work though.
Personally, if I was dead set with your $1000 budget, I would install the 1.5" body lift and the Cognito UCAs, then decrank your leveling keys to where it had just enough lift for the tires you want to run.
From there, there aren't any good ~3" lifts out there. If you had $2000 to spend, I would go for something like the 4" Cognito lift, and remove your leveling keys and put the stock ones back in. You could run it decranked at 3-4" of lift, have a nice soft ride, stable for work and towing, flat suspension/steering/drive angles that will make for longer lasting and stronger components, and not have the ugly frame hanging down of a body lift. This is the ideal option, and will make you the happiest in the long run.
Another option, IMO Rough Country makes probably the best budget 6", not the best, not the cheapest, but the best value IMO, for ~$1200. Again, you will want to ditch the leveling keys, and go back to the stock keys. Run it decranked if you wish to around 4-5" for a nice ride and flat angles, same benefits as above, just a little higher up.
Since you mentioned you are running 17" tires...I assume you are running wider aftermarket 17" wheels instead of the stock 16s. This is probably the cause of much of your rubbing. On stock, or stock offset wheels, you can easily fit up to 35x12.5s with minimal cranking and trimming. I ran this setup for several years on a couple of my 2500HDs. You could look into a set of Hummer H2 or 07.5+ 2500HD/3500 17" wheels, both are 17" and have the right width and offset to fit larger tires without many modifications.
There are a number of modifications you can do to increase clearance, such as raising the front bumper 2" (look at 2" body lift bumper brackets for inspiration), then you can use a thinner 1500 top plastic bumper cap to replace the tall 2500HD bumper, IMO it looks better without the giant piece of plastic, and it gives you 2" more clearance at the tire. Another common place to rub is the drivers side inner wheel well, behind the tire. You may notice a bulge there where the parking brake cable comes out of the cab, you can use a couple nice big zip ties and pull that cable in closer to the body mount, then take a heat gun, heat up the inner fender plastic and push it back a bit (you can also pop a couple holes and zip tie it to hold it back as well if you wish). If you really wanted to get into it, you can remove the fender flare, clearance a bit behind it, then put it back on, heat it up a bit and push it back into place, hiding all evidence behind it.
As far as tires, I second Coopers, the STT is a great tire and decently affordable, another option is their new ST Maxx, it is still aggressive looking on the outer tread lugs, but the center tread is a little more closely spaced like an all terrain, it should be more highway friendly, last longer, etc. Nitto Trail Grapplers are great as well. As mentioned, you don't want to go too tall, since you tow a lot as it gives a higher final gear ratio, losing you precious mechanical advantage when towing.
Personally, given your towing needs, and budget, I would forget the lift, decrank a bit, and go back to some nice 265, maybe 285 size tires that will fill out the fender well plenty, get you where you need to go, and be much friendlier when towing as well as on your fuel bill.
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