Parameters and Secrets Lambda Extension with SnapStart

Lambdas running with SnapStart do not provide the AWS_SESSION_TOKEN environment variable. Instead, they use container credentials. However, when I tried the following, I received a "bad session token or header key" return value.

ContainerCredentialsProvider containerCredentialsProvider = ContainerCredentialsProvider.builder().build();
AwsCredentials awsCredentials = containerCredentialsProvider.resolveCredentials();

if (awsCredentials instanceof AwsSessionCredentials awsSessionCredentials) {
        HttpResponse<String> response = httpClient.send(HttpRequest.newBuilder()
            .GET()
            .uri(URI.create("http://localhost:2773" + url))
            .header("X-Aws-Parameters-Secrets-Token", awsSessionCredentials.sessionToken())
            .build(), HttpResponse.BodyHandlers.ofString());
        return response.body();
} else {
    throw new RuntimeException("Invalid environment");
}

So how can I use the extension with SnapStart if the above does not work?