Here is my solution.
Setup
File structure:
app  
  |--src
    |--assets
      |--images
        |--logos
          |--small_kl_logo.png
          |--small_a1_logo.png
          |--small_kc_logo.png
          |--small_nv_logo.png
          |--small_other_logo.png
        |--index.js
    |--SearchableList.js
In index.js, I have this:
const images = {
  logos: {
    kl: require('./logos/small_kl_logo.png'),
    a1: require('./logos/small_a1_logo.png'),
    kc: require('./logos/small_kc_logo.png'),
    nv: require('./logos/small_nv_logo.png'),
    other: require('./logos/small_other_logo.png'),
  }
};
export default images;
In my SearchableList.js component, I then imported the Images component like this:
import Images from './assets/images';
I then created a new function imageSelect in my component:
imageSelect = network => {
  if (network === null) {
    return Images.logos.other;
  }
  const networkArray = {
    'KL': Images.logos.kl,
    'A1': Images.logos.a1,
    'KC': Images.logos.kc,
    'NV': Images.logos.nv,
    'Other': Images.logos.other,
  };
  return networkArray[network];
};
Then in my components render function I call this new imageSelect function to dynamically assign the desired Image based on the value in the this.state.network:
render() {
  <Image source={this.imageSelect(this.state.network)} />
}
The value passed into the imageSelect function could be any dynamic string. I just chose to have it set in the state first and then passed in.
I hope this answer helps. :)