Out of those it would be the LP for me as I have the custom and its a great axe, only problem is that you would be a bit limited with the sounds so it would play rock and blues and obviously some not so distorted metal well. Anything else you would be a bit limited on, strat's are versatile but unique at the same time. I do not think I would go for the delux players,
have a look at the american delux. That has the S-1 switch for similar muddy tones to a LP and is a truely great axe. They retail at around $1100 so just out your budget but you will not need any other guitars....Look at the reviews here:
Buy Fender American Deluxe Strat V-Neck Electric Guitar at Musician's Friend