TypeScript, Serverless and Webpack complications

Basically it doesn't matter either it is serverless application or not. What matters is your webpack config. Try using serverless-webpack plugin with proper webpack config for typescript compilation

serverless.yml webpack part example:

plugins:
  - serverless-webpack
custom:
  webpack:
    webpackConfig: ./webpack.config.js

webpack.config.js example:

const path = require('path')
const slsw = require('serverless-webpack')
module.exports = {
  mode: 'development',
  devtool: 'source-map',
  entry: slsw.lib.entries,
  target: 'node',
  resolve: {
    extensions: ['.mjs', '.ts', '.js', '.json', '.tsx']
  },
  output: {
    libraryTarget: 'commonjs2',
    path: path.join(__dirname, '.webpack'),
    filename: '[name].js'
  },
  module: {
    rules: [
      {
        oneOf: [
          {
            test: /\.(ts|js)$/,
            exclude: [/node_modules/],
            loader: 'ts-loader',
            options: {
              transpileOnly: true
            }
          }
        ]
      }
    ]
  }
}

You can adapt it to your needs, I've just shared a little piece of my config