I have tried using the hooks below;

<?php ba_show_banner(1); ?>

and

<?php if(function_exists('ba_hook')) { ba_hook('hook1'); } ?>

None is displaying the Banner in the blog, it only works on the main website.

Is there a specific hook to use in the blog plugin to show the banners?

Follow this

https://osclasspoint.com/blog/advertisement-and-banners-b1

Blog plugin does not support banner ads plugin.

I wanted to purchase the Advertisement Manager

But I read "We recommend to use Banner Ads Plugin instead, that is connected to other plugins, provides modern solution for advertisement management and is updated more frequently." before I purchased the banner ads plugin.

Now you're telling me it doesn't work with the Blog plugin.

Isn't that swindling?

Only some themes allow to put banners/adverts from Banner Ads Plugin into textarea for banner.
Advertisement manager does not work with Blog plugin as well.
Means, in blog plugin, you must define your banners manually as HTML or you will modify plugin files and place code from Banner Ads Plugin.

Btw. there is no mention anywhere that blog plugin would be integrated with banner ads plugin.