If I export this crop wouldn't it resize the small crop to the overall image size I want, making the entire image dozens of times larger than I had hoped for...?
Why would it do that? Let's say your image is 4000x6000 pixels. You want to print it on 30 x 45 inch paper at 300 ppi. That means that LR will have to enlarge it to 9000x13500, a 2.25X enlargement. If you crop out an 800x1200 section and set LR to enlarge it by 2.25X to 1800x2700 pixels, that will give you 6 x 9 inch, 300 ppi print. Print it on 8 x 10 paper with 1 inch margins on the long side and 0.5 inch margins on the short side, or better yet, send it to the same lab where you plan to do the big print and instruct him to print it "as is".