Excel JS Addin from existing web service

I am trying to develop an Excel add-in with custom functions and a task pane that interacts with an existing remote web application, to which I have added pages that load OfficeJS and my custom functions. The service uses cookie-based session authentication after login. What's the simplest way to get authentication working in the fetch requests used in my custom formulas? Do I have to use the Office Dialog API for auth? I can do local sign-in in the Taskpane directly - why would I need to use the Office Dialog API?

Once I have a valid authenticated browser session, can I use that session in a shared runtime to automatically authenticate the ajax requests made by my custom functions? I have tried to get a shared runtime going to test this, to no avail. My manifest is valid, the Add-in will install via sideload, and the task pane will load, but when I execute a custom function the shared runtime tries to start and then times out, with no error message in the runtime log.

Per conversation in comments, to use cookie based authentication for Excel add-ins or custom functions, you need to use the Shared Runtime. You can configure by adding the Runtimes element into your manifest, and use the same resource ID/page for all the components you want shared, which in this case would be your taskpane and your custom function.

<VersionOverrides xmlns="http://schemas.microsoft.com/office/taskpaneappversionoverrides" xsi:type="VersionOverridesV1_0">
  <Hosts>
    <Host xsi:type="Workbook">
      <Runtimes>
        <Runtime resid="ContosoAddin.Url" lifetime="long" />
      </Runtimes>
      ...

      <Page>
          <SourceLocation resid="ContosoAddin.Url"/>
      </Page>
...

A detailed walk through can be found here: https://learn.microsoft.com/en-us/office/dev/add-ins/tutorials/share-data-and-events-between-custom-functions-and-the-task-pane-tutorial