I was 90% done migrating to flexible env when I came across this same problem. After extensive research, I concluded there are three options:
REST API (experimental)
Use the beta REST API for task queues (this, as all other google APIs from flexible env, is external, so you need to deal with auth appropriately).
REST API reference: https://cloud.google.com/appengine/docs/python/taskqueue/rest/
Note, this is external and experimental. Find e.g. a java sdk without any meaningful documentation here: https://developers.google.com/api-client-library/java/apis/ (current version: https://developers.google.com/api-client-library/java/apis/taskqueue/v1beta2)
Compat runtime
Build your own flexible environment, based off a -compat runtime. This offers the old appengine api in a container suitable for the flexible env:
https://cloud.google.com/appengine/docs/flexible/custom-runtimes/build (look for images with "YES" in the last column)
e.g.: https://cloud.google.com/appengine/docs/flexible/java/dev-jetty9-and-apis
https://cloud.google.com/appengine/docs/flexible/java/migrating-an-existing-app
Note: I spent two weeks in blistered frustration pleading every God almighty help me get this to work, following container rabbit holes into the depths of Lucifer's soul and across unexplored dimensions. I eventually had to give in. I just can't get this to work to a satisfying degree.
Proxy service
Kind of a hacky alternative, but it gets the job done: create a very thin standard environment wrapper service which proxies tasks onto / off your queue. Pass them to your own app however you want. ¯\_(ツ)_/¯
Downside is you are now spinning up extra instances and burning extra minutes.
I ended up with a variation of this, where I'm using a proxy service in standard env, but just ported my eventual task handler to AWS Lambda (so it's completely off GAE). It's a different disaster, but a more manageable one.
Good luck!