I paid for subscription for Britbox as I wanted to watch blue lights season 2 but I still cannot access the season without an additional charge of $2.00 an episode when I have been charged $10.99 for a month
If you see a mention of a price per episode, then that is not the subscription, but the Store buy version.
10.99/m matches the pricing for Canada. Are you in Canada?
From the main page for the show, find the episode list, then click the description (not the thumbnail) for S02E01 in the Apple TV app. On the next page (details page for that episode), scroll down to How to Watch, then select your preferred content provider (i.e. how you subscribed), to start watching through there.
This should update the Continue Watching queue list with the wanted content provider link, so that subsequent episodes are more easily selectable.