I think I found half-way solution, which should work in my case. We use web view only for login, once cooke is obtained and put into secure storage we discard webview. So in that case before get rid of webview, what I need to do is to clear cookies and save the empty jar to the disk. In my case of fandroid application I missed just one step:

await CookieManager.clearAll();
await CookieManager.flush();

all other parameters for WebView like cacheMode and incognitoMode etc do not matter.