How can I send one bit one mile with one battery?
Amateur radio is perfect for that. Hand-held radios can reach several miles. You can use voice or morse if necessary. There is rarely any congestion, there may even be some repeaters. The initial test is pretty easy to pass as long as you get the manual.

Radios: http://www.aesham.com/ham-radios-handheld/?p=catalog&amp;mode=catalog&amp;parent=274&amp;pg=1&amp;CatalogSetSortBy=price

Study Guides: http://www.arrl.org/ham-radio-license-manual