Hi @Michael McNicholas 
I'm trying the exact same.
We use the Amplitude Analytics Browser SDK” GTM community template, and now want to also use Experiment via a Custom HTML. I have tried with this code, that are loaded after Amplitude Browser SDK. 
(Got some help from ChatGPT...)

<script src="https://unpkg.com/@amplitude/experiment-js-client@1.9.0/dist/experiment.umd.js"></script>
<script>
  // Initialize the experiment client with your deployment key
  var experiment = Experiment.Experiment.initializeWithAmplitudeAnalytics('YOUR_DEPLOYMENT_KEY');

  // Start the SDK and wait for it to be ready
  experiment.start().then(function() {
    // Once the SDK is ready, check the feature flag variant
    var variant = experiment.variant('YOUR_FLAG_KEY');
    if (variant.value === 'on') {
      // Code to execute if the flag is on
    } else {
      // Code to execute if the flag is off
    }
  });
</script>

I have now tried this. The script is loaded but i get a 401, and the consol says “Missing Api-Key in Authorization header”. 
I have double checked the deployment API-key, and the flag_key.

Would be interesting to also hear from others what the best practice is for GTM implementation.

What ended up working for me… is to chain the GTM calls - i.e. “Fire a tag after this one fires” 

So the Amplitude Analytics Browser SD fires in one tag,

then the Experiment ones fires in a second tag (calling https://unpkg.com/@amplitude/experiment-js-client@1.9.0/dist/experiment.umd.js) 

Then I initalize the initialize the experiment object (since I know the previous tag has fired) 

experiment = Experiment.Experiment.initializeWithAmplitudeAnalytics('YOUR_DEPLOYMENT_KEY');

in a third tag

then I run the code for a specific experiment, get variations, etc in the fourth tag. And I am sure each part of the experiment code only runs when the part before it has been called. 

…Otherwise, there might be a race condition where the second and later lines in the tag are running before the first part is complete.  Hope this helps. 

Here’s a custom HTML tag that works well. Use the “Initialization - All Pages” trigger for it. Then you can use the code in the comment at the top in other custom HTML tags to retrieve variants.

<script>
  /**
   * Amplitude Experiment Google Tag Manager: Custom HTML Script
   * -----------------------------------------------------------
   *
   * This script loads and starts the Experiment SDK, then sets window variables
   * after the script has loaded:
   *
   *   - `window.experiment`: The Experiment SDK instance. May be undefined.
   *
   *   - `window.experimentReady`: A promise which resolves with the experiment
   *        SDK instance when the Experiment SDK has completed starting and is
   *        ready to be used.
   *
   * To use the Experiment SDK outside the tag, wait for the ready promise to
   * resolve and use the SDK directly to access variants for the user:
   *
   * ```
   * window.experimentReady.then(function (experiment) {
   *   var variant = experiment.variant("my-experiment");
   *   if (variant.value === "treatment") {
   *     // Treatment
   *   } else {
    *    // Control
   *   }
   * })
   * ```
   *
   * Note: this script assumes you have are using the Amplitude Analytics SDK
   * to send events.
   */

  // TODO: Set your Amplitude Experiment deployment key.
  var deploymentKey = "DEPLOYMENT_KEY_HERE";

  // TODO (optional): Configure the Experiment SDK.
  var config = {};

  // SDK version
  var version = "1.9.1"

  window.experimentReady = new Promise(function(resolve) {
    function loadAsync(src, callback) {
      var script = document.createElement('script');
      script.src = src;
      if (script.readyState) { // IE, incl. IE9
        script.onreadystatechange = function() {
            if (script.readyState === "loaded" || script.readyState === "complete") {
                script.onreadystatechange = null;
                callback();
            }
        };
      } else {
        script.onload = function() { // Other browsers
            callback();
        };
      }
      document.getElementsByTagName('head')h0].appendChild(script);
    }

    function experimentSDK() {
      return "https://unpkg.com/@amplitude/experiment-js-client@" + version + "/dist/experiment.umd.js"
    };

    loadAsync(experimentSDK(), function () {
        window.experiment = Experiment.Experiment.initializeWithAmplitudeAnalytics(deploymentKey, config);
        window.experiment.start().then(function () {
          window.experiment.variant("FLAG_KEY_HERE"); // If you have an immediate variant to retrieve
          resolve(window.experiment);
        });
    });
  });
</script>