I have a select2 drop-down where you need to first enter 2 characters and then select your item. I am unable to test this with Protractor.
var select2 = element(by.css('div#s2id_person'));
select2.click();
select2.sendKeys('ip');
select2.sendKeys(protractor.Key.ENTER);
The following gets an error about being unable to focus the element when you try to sendKeys.
The following snippet successfully activates and selects the first option in a select2 widget, it allows for loading options over a network.
There are several issues with the 'select2' widget - in regard to protractor E2E testing - which this snippet addresses. The comments explains it all very well.
In your provided scenario, you would use it like this: