Using requests to post to a website after simulating an OnClick event without a headless browser

I have several constraints due to packages and software being locked down in my company.

I am working with a website that has an onClick event that reveals a table. Before the onClick event (i.e. the initial page of the website), I can write a POST request that essentially mimics the action of the onClick event, returning the page post the onClick event. However, additional POST requests or GET requests, even within the same session,
are interacting against the original page. I have not “clicked” the button to reveal the JavaScript that the initial POST request reveals in the requests session. The click event does not change the URL (so there are no parameters to load against; it just loads the HTML directly onto the page).

My question is: How would I resolve this without using a headless browser or changing the website (and adding something like AJAX). My idea was to use the content from the initial response as the “website” I POST against, but I don’t how to make that work (or if it can work).

Some simplified code below to describe the situation:

import requests
url = "someurl.com"

session = requests.Session()
headers = {"some keys and values"}
payload1 = {"submitBtn": "submit"}
initialRequest = session.post(url, data=payload1, headers=headers)
// initialRequest.content -> This returns the "correct" page (the page the javascript loads after the onClick event)
payload2 = {"newData": "3", "saveBtn":"save"}
followUpRequest =  session.post(url, data=payload2, headers=headers) // This fails because Requests is loading the initial page again (demonstrated by a session.get(url) Request that returns the initial page).

Is this a way to POST against the content from “initialRequest” on the same URL? Again the URL does not change at all (so there’s no easy parameters I can throw in). Any ideas? Other mainstream python libraries would work but Selenium is not an option.