Hi Tyrone
It seems like something in a plugin or theme is interfering with the pagination.
WP has several hook which can be used to modify the html of the actual pagination, which I think is happenign on your setup.
Probably there is a check and if is_single() is true then something is removing the pagination.
When an ajax request is performed is_single will be false which might be why you are seeing this.
I would try changing your theme, and disabling your plugins to find the cause.
Alternatively, you could try to create your pagination in another way, something where this other plugin or theme won’t modify the pagination – perhaps you can try this – https://wordpress.org/plugins/wp-pagenavi/ – The S&F results template has some sample code for this plugin.
Thanks