I spent a good deal of time working on this same thing and eventually found a solution using the requests library that seems to work well. It even handles setting multiple cookies in one response, which took a bit of investigation to figure out. Here's the flask view function:
from dotenv import load_dotenv  # pip package python-dotenv
import os
#
from flask import request, Response
import requests  # pip package requests
load_dotenv()
API_HOST = os.environ.get('API_HOST'); assert API_HOST, 'Envvar API_HOST is required'
@api.route('/', defaults={'path': ''})  # ref. https://medium.com/@zwork101/making-a-flask-proxy-server-online-in-10-lines-of-code-44b8721bca6
@api.route('/<path>')
def redirect_to_API_HOST(path):  #NOTE var :path will be unused as all path we need will be read from :request ie from flask import request
    res = requests.request(  # ref. https://stackoverflow.com/a/36601467/248616
        method          = request.method,
        url             = request.url.replace(request.host_url, f'{API_HOST}/'),
        headers         = {k:v for k,v in request.headers if k.lower() != 'host'}, # exclude 'host' header
        data            = request.get_data(),
        cookies         = request.cookies,
        allow_redirects = False,
    )
    #region exlcude some keys in :res response
    excluded_headers = ['content-encoding', 'content-length', 'transfer-encoding', 'connection']  #NOTE we here exclude all "hop-by-hop headers" defined by RFC 2616 section 13.5.1 ref. https://www.rfc-editor.org/rfc/rfc2616#section-13.5.1
    headers          = [
        (k,v) for k,v in res.raw.headers.items()
        if k.lower() not in excluded_headers
    ]
    #endregion exlcude some keys in :res response
    response = Response(res.content, res.status_code, headers)
    return response
Update April 2021: excluded_headers should probably include all "hop-by-hop headers" defined by RFC 2616 section 13.5.1.