Wow nice job on the church site using the cleanout template!
There isn't a way to do a specific background for specific pages in Vertex so yes you would need to do the module method. To do so full width you would need to open up your index.php file and remove the class s5_wrap as this will constrain the width of this position:
<div id="s5_top_row1_area_inner" class="s5_wrap">
Then you can add any HTML/CSS you need and publish to any of the top row 1 positions and it will get the full width left to right.
Let me know if you need any further help