This doesn't really have anything to with async functions specially. If you want to call a function internally and export it, define it first and then export it.

async function doStuff() {
  // ...
}
// doStuff is defined inside the module so we can call it wherever we want

// Export it to make it available outside
module.exports.doStuff = doStuff;

Explanation of the problems with your attempts:

module.exports.PrintNearestStore = async function PrintNearestStore(session, lat, lon) {
...
}

This does not define a function in the module. The function definition is a function expression. The name of a function expression only creates a variable inside the function itself. Simpler example:

var foo = function bar() {
  console.log(typeof bar); // 'function' - works
};
foo();
console.log(typeof foo); // 'function' - works
console.log(typeof bar); // 'undefined' - there is no such variable `bar`

See also Named function expressions demystified. You could of course refer to the function if you'd refer to module.exports.PrintNearestStore everywhere.

module.exports.PrintNearestStore = PrintNearestStore;

var PrintNearestStore = async function(session, lat, lon) {

}

This is almost OK. The problem is that the value of PrintNearestStore is undefined when you assign it to module.exports.PrintNearestStore. The order of execution is:

var PrintNearestStore; // `undefined` by default
// still `undefined`, hence `module.exports.PrintNearestStore` is `undefined`
module.exports.PrintNearestStore = PrintNearestStore;

PrintNearestStore = async function(session, lat, lon) {}
// now has a function as value, but it's too late

Simpler example:

var foo = bar;
console.log(foo, bar); // logs `undefined`, `undefined` because `bar` is `undefined`
var bar = 21;
console.log(foo, bar); // logs `undefined`, `21`

If you changed the order it would work as expected.

module.exports.PrintNearestStore = async function(session, lat, lon) {
    await PrintNearestStore(session, lat, lon);
}

var PrintNearestStore = async function(session, lat, lon) {
...
}

This works because by the time the function assigned to module.exports.PrintNearestStore is executed, PrintNearestStore has the function as its value.

Simpler example:

var foo = function() {
  console.log(bar);
};
foo(); // logs `undefined`
var bar = 21;
foo(); // logs `21`

Error with first case: PrintNearestStore - Function expression, so this name not available outside.

error with second case: using variable, instead Function declaration. In this case, declaration of variable PrintNearestStore are hoisted, so, you can use this name before line var PrintNearestStore = ..., but in this case value would be undefined.

So, simplest solution change second variant like this:

module.exports.PrintNearestStore = PrintNearestStore;

async function PrintNearestStore(session, lat, lon) {

}

an alternative would be to export like this. // foo.js

export async function foo(){ 
 console.log('I am greatest of all.'); // for the person who reads it, just say it.
}

then use it in other scripts like

import { foo } from './foo'

foo();