Solution 1:

So, after a few days of struggling, I have settled with the following solution, which I admit is slighlty obscure, but works perfectly (incl. back button, gestures):

In the React Web Application, I keep the ReactRouter history object available through the global Window object.

import { useHistory } from 'react-router'
// ...

declare global {
  interface Window {
    ReactRouterHistory: ReturnType<typeof useHistory>
  }
}

const AppContextProvider = props => {
  const history = useHistory()

  // provide history object globally for use in Mobile Application
  window.ReactRouterHistory = history

  // ...
}

In the React Native Mobile Application, I have custom code injected to the WebView, that makes use of the history object for navigation and the application communicates with this code using messages:

webViewScript.ts

// ...
export default `
  const handleMessage = (event) => {
    var message = JSON.parse(event.data)
    switch (message.type) {
      // ...
      case '${MessageTypes.NAVIGATE}':
        if (message.params.uri && message.params.uri.match('${APP_URL}') && window.ReactRouterHistory) {
          const url = message.params.uri.replace('${APP_URL}', '')
          window.ReactRouterHistory.push(url)
        }
        break
    }
  };

  !(() => {
    function sendMessage(type, params) {
      var message = JSON.stringify({type: type, params: params})
      window.ReactNativeWebView.postMessage(message)
    }

    if (!window.appListenersAdded) {
      window.appListenersAdded = true;
    
      window.addEventListener('message', handleMessage)
      
      var originalPushState = window.history.pushState
      window.history.pushState = function () {
        sendMessage('${MessageTypes.PUSH_STATE}', {state: arguments[0], title: arguments[1], url: arguments[2]})
        originalPushState.apply(this, arguments)
      }

    }
  })()
  true
`

intercom.ts (no routing specifics here, just for generic communication)

import WebView, {WebViewMessageEvent} from 'react-native-webview'
import MessageTypes from './messageTypes'


export const sendMessage = (webview: WebView | null, type: MessageTypes, params: Record<string, unknown> = {}) => {
  const src = `
  window.postMessage('${JSON.stringify({type, params})}', '*')
  true // Might fail silently per documentation
  `
  if (webview) webview.injectJavaScript(src)
}

export type Message = {
  type?: MessageTypes,
  params?: Record<string, unknown>
}

export type MessageHandler = (message: Message) => void

export const handleMessage = (handlers: Partial<Record<MessageTypes, MessageHandler>>) => (
  (event: WebViewMessageEvent) => {
    const message = JSON.parse(event.nativeEvent.data) as Message
    const messageType = message.type

    if (!messageType) return
    const handler = handlers[messageType]
  
    if (handler) handler(message)
  }
)

export {default as script} from './webViewScript'

WebViewScreen.tsx

import {handleMessage, Message, script, sendMessage} from '../intercom'
// ...

const WebViewScreen = ({navigation, navigationStack}: WebViewScreenProps) => {
  const messageHandlers = {
    [MessageTypes.PUSH_STATE]: ({params}: Message) => {
      if (!params) return
      const {url} = params
      const fullUrl = `${APP_URL}${url}`
      navigationStack.push(fullUrl)
    },
    // ...
  }

  const uri = navigationStack.currentPath

  // navigation solution using history object propagated through window object
  useEffect(() => {
    if (uri) {
      sendMessage(webViewRef.current, MessageTypes.NAVIGATE, {uri})
    }
  }, [uri])

  // this is correct! Source is never going to be updated, navigation is done using native history object, see useEffect above
  // eslint-disable-next-line react-hooks/exhaustive-deps
  const source = useMemo(() => ({uri}), [])

  return (
    <View
        // ...
        refreshControl={
          <RefreshControl
              // ...
          />
        }
    >
      <WebView
          source={source}
          injectedJavaScript={script}
          onMessage={handleMessage(messageHandlers)}
          // ...
      />
    </View>
  )

}