You've done all of the usual things, so it's difficult to understand what other factor could be relevant here. Two thoughts:

(1) On the problem sites, is a right-click or a Shift+right-click recognized when a regular click isn't? If you get a context menu, does it have the normal items for links or does it show the more general menu for a blank area of a page?

(2) If it is enabled on the Settings page, could you disable the "Automatically refuse cookie banners" setting.

On both sites right click brings up the context menu with Save Page as/Save All/Take Screenshot/etc If I click on the search Box I get Undo/Redo/Cut/Copy/Paste/etc

Bunnings has a notification on the top. If I mouse over the X on the top right the mouse point turns to a hand but I can't close the notification If I type in the search boxes, what I type shows (and search history or suggestions are listed) but I can't submit the search.

Where is the setting for "Automatically refuse cookie banners"? I can't seem to find it.

It's rare, but sometimes Firefox doesn't get the correct mouse coordinates. If you try clicking around the search button a quarter inch in all directions, does anything happen? And just to make sure search is working at all, can you submit the query using the Enter key?

For the site where what you right-clicked is not a link, it could be that coordinate issue, or it could be that something transparent is in front of the link. If you right-click > Inspect, Firefox should open the Page Inspector tool. This displays the HTML of the page in a tree diagram. Normally, when you right-click > Inspect a link, the element highlighted is either an "a" element (anchor element) or some content nested under the "a" element. You can compare on simple pages to see what I mean. What kind of element is getting highlighted on the Bunnings site?

For cookie banners, try using the tiny search box in the page to search for banner (or maybe it didn't exist yet on Firefox 140).