For under $1000 you can get the Sigma 120-400 OS. Or, for just over $1000, the Sigma 150-500 OS. Those two would be near the top of my shopping list, if it were me.
Assuming most lacrosse is played during daytime, with reasonably good light, these would be fine on your 50D. They don't work well with teleconverters, but you will have long enough focal length in the lens alone that a TC shouldn't be needed.
The Canon 100-400 IS is another lens that would work well, but normally sells for a bit more than your authorized budget. It is currently on sale for $1450, though... so wouldn't stretch your wallet too much. This is unique in that it's a push/pull zoom, which some people love, others of us really don't like. It's fast operating, but personally I always found it more difficult to get a steady shot with a push/pull zoom. The IS offered with this lens might help with that, though. The Canon 100-400 is an older design now... both the Sigmas above are much newer and supposedly have better stabilization. As to image quality, IMO the Sigmas are a little lower contrast and give slightly lower color saturation... but both of those are adjustable in post processing. The Canon is a little more compact than the Sigmas.
The Canon 400/5.6 is within your budget, and would likely be the sharpest possible choice, but doesn't have IS, so you might want a tripod or at least a monopod. To be fair, you might want one anyway since most of these lenses can get pretty heavy after a while. Even on a tripod or monopod, and shooting sports where you may want to use higher shutter speeds to freeze the action, IS can be very nice to have. It will definitely make possible some shots you might not get otherwise (note: Sigma's OS is essentially the same as Canon's IS). Perhaps more importantly, this prime lens only gives you a single focal length to work with, doesn't have near the versatility of a zoom, which can be very handy when tracking subjects at various distances around a playing field.
For handheld shooting I use a Canon 300/4 IS with and without a Canon 1.4X II. This gives me two very usable focal lengths. Bought new, the lens along is a bit more than your budget, but if you considered used you can find one for considerably less (I paid $900 for mine, used but "like new"). The teleconverter adds another $250 or so cost. I've recently seen info about the new Kenko MC4 1.4X DGX that makes that TC, at $150, appear to be a very good alternative. Lenses might not focus as quickly with a third party TC, though. I haven't tried it, so can't say for certain.
The Canon EF-S 55-250 IS which you already have is - as I'm sure you know - quite inexpensive, compact, gives darned good image quality for the money and has IS, but doesn't have fast/accurate USM focus that can be important tracking and shooting sports/action and really doesn't increase your reach very much over the 80-200 you already have.
There are several Canon 70-300mm models. I don't use any of these, but would suggest at least the latest version with IS and USM that sells for about $650 (not the cheaper 75-300 without USM). There is also a 70-300 IS USM DO, a very compact lens... I know one sports photographer using one but haven't used it myself. The DO costs close to your budget.
It's a lot larger, but quite good optically, the Canon 75-300mm "L" IS USM is currently on sale for right at your budget: $1400. There are some sports photographers using it, too. Normally it's a $1600 lens.
A more recent 70-200/4 IS plus a 1.4X teleconverter also might be within your budget (the lens is currently on sale). Great lens, a real workhorse. But this combo only gets you to an effective 280mm, so isn't a very big improvement over your current setup.
Pretty much any of the above is going to give you better IQ than you are seeing with your 80-200 and 1.4X combo. I suspect most of them will give you faster and more accurate focus, too. The only zooms Canon recommends using with a teleconverter are their 70-200s.
In order to use a 2X teleconverter with any 70-200mm, to get to an effective 400mm, you basically would need to go to an f2.8 lens which are bigger, heavier and more expensive. With most combos the hit to image quality is too great to make them worthwhile. The Canon 70-200/2.8 IS "Mark II" and the Canon 2X "Mark III" do work together a lot better than most. But the cost for this particular lens and teleconverter combo is way, way more than your budget.