How about changing the Paper size in the Printer dialog. LR only know what the paper size is by looking at the Printer options that is selected.
You will have to go into the page set up, which opens your printer driver for your Lexmark printer. Choose 5 x 7" paper and borderless printing. Set the proper orientation. Then, in Lightroom, you'll probably have to experiment with cell size and borders to get it to be a borderless print. But the paper size is set in the page setup options. After you get all that done then you can print to JPEG and have the file ready to go to Costco.
This all seems like a lot of work for something that should be much easier. There's no need to muck about with the Print module for this.
if you've already cropped to 5x7 aspect ratio, just click the "Export" button, or File > Export.
1. Make sure you choose to export to Hard Drive.
2. Change the the file type to JPG. Also, change the Color space to your Costco profile if desired.
3. Set quality to 100 for best results
4. Disable "image sizing."
5. Sharpen as appropriate.
Than an click export at the bottom, save your file and upload to Costco.