SecretJSを使用したNodeJSのソリューション
AWSがNodeJSで公開されないため、秘密鍵をSDKから削除したのはばかげているようです。
私はNodeJSでフェッチをインターセプトし、@ Simon Buchanの回答を使用してハッシュされたキーを追加することでそれを機能させました。
cognito.js
import { CognitoUserPool, CognitoUserAttribute, CognitoUser } from 'amazon-cognito-identity-js'
import crypto from 'crypto'
import * as fetchIntercept from './fetch-intercept'
const COGNITO_SECRET_HASH_API = [
  'AWSCognitoIdentityProviderService.ConfirmForgotPassword',
  'AWSCognitoIdentityProviderService.ConfirmSignUp',
  'AWSCognitoIdentityProviderService.ForgotPassword',
  'AWSCognitoIdentityProviderService.ResendConfirmationCode',
  'AWSCognitoIdentityProviderService.SignUp',
]
const CLIENT_ID = 'xxx'
const CLIENT_SECRET = 'xxx'
const USER_POOL_ID = 'xxx'
const hashSecret = (clientSecret, username, clientId) => crypto.createHmac('SHA256', clientSecret)
  .update(username + clientId)
  .digest('base64')
fetchIntercept.register({
  request(url, config) {
    const { headers } = config
    if (headers && COGNITO_SECRET_HASH_API.includes(headers['X-Amz-Target'])) {
      const body = JSON.parse(config.body)
      const { ClientId: clientId, Username: username } = body
      // eslint-disable-next-line no-param-reassign
      config.body = JSON.stringify({
        ...body,
        SecretHash: hashSecret(CLIENT_SECRET, username, clientId),
      })
    }
    return [url, config]
  },
})
const userPool = new CognitoUserPool({
  UserPoolId: USER_POOL_ID,
  ClientId: CLIENT_ID,
})
const register = ({ email, password, mobileNumber }) => {
  const dataEmail = { Name: 'email', Value: email }
  const dataPhoneNumber = { Name: 'phone_number', Value: mobileNumber }
  const attributeList = [
    new CognitoUserAttribute(dataEmail),
    new CognitoUserAttribute(dataPhoneNumber),
  ]
  return userPool.signUp(email, password, attributeList, null, (err, result) => {
    if (err) {
      console.log((err.message || JSON.stringify(err)))
      return
    }
    const cognitoUser = result.user
    console.log(`user name is ${cognitoUser.getUsername()}`)
  })
}
export {
  register,
}
fetch-inceptor.js(https://github.com/werk85/fetch-intercept/blob/develop/src/index.jsの ForkからNodeJSにフォークして編集)
let interceptors = []
if (!global.fetch) {
  try {
    // eslint-disable-next-line global-require
    global.fetch = require('node-fetch')
  } catch (err) {
    throw Error('No fetch available. Unable to register fetch-intercept')
  }
}
global.fetch = (function (fetch) {
  return (...args) => interceptor(fetch, ...args)
}(global.fetch))
const interceptor = (fetch, ...args) => {
  const reversedInterceptors = interceptors.reduce((array, _interceptor) => [_interceptor].concat(array), [])
  let promise = Promise.resolve(args)
  // Register request interceptors
  reversedInterceptors.forEach(({ request, requestError }) => {
    if (request || requestError) {
      promise = promise.then(_args => request(..._args), requestError)
    }
  })
  // Register fetch call
  promise = promise.then(_args => fetch(..._args))
  // Register response interceptors
  reversedInterceptors.forEach(({ response, responseError }) => {
    if (response || responseError) {
      promise = promise.then(response, responseError)
    }
  })
  return promise
}
const register = (_interceptor) => {
  interceptors.push(_interceptor)
  return () => {
    const index = interceptors.indexOf(_interceptor)
    if (index >= 0) {
      interceptors.splice(index, 1)
    }
  }
}
const clear = () => {
  interceptors = []
}
export {
  register,
  clear,
}