It may require a request of the (commercial?) printer to "FIT" the image on the 8x10 page. Ask how their systems work!
For your own inkjet printer, the option in print module would be to NOT "Zoom to Fill"
Your image should be cropped in the 7:10 ratio format when you export from Lightroom.
In the (Develop) Crop tool module you may already have a 7x10 option, or choose "Enter Custom" and fill the dialog-
In my experience I have had 15x20 images printed on 16x20 paper, and the result was the image with a 1inch white border on one side ( to trim off)
I'm working with a pro printer, they don't have Lr in house so they were unable to provide directions. I have mastered the 7:10 crop but am stuck on how to export so that my 7:10 crop will be printed on 8:10 paper.
There is nothing special you need to do to get a 7x10 printed on the 8x10 paper.
Just export your 7x10 image as a JPG or TIF using the resolution and color profile recommended by the printer.
When they print your image, it will either be centered with 1/2 inch margins on each side, or have a one inch margin all on one side. The printer need to just make sure they don't "stretch to fit" to use all the paper.