Content vs Template on custom post

You can not replace half of a template by using filters and hooks if the template itself doesn’t support so.

The template_include will override an entire template, it means header, body, content, sidebar, footer, and everything.

Now you have these options:

Style Your Template

Create a full single-my_member.php file and then style it by CSS and probably JS. You need to think of it as a complete template, not just a page to render a few lines of content.

Use preg_replace()

Not a good idea at all, but you could do a preg_replace() combined with an output buffering and add content to a page’s common places, such as <body> or <header>. This might slow down your site a lot, and is not always accurate.

Provide Shortcodes

If you are trying to add limited content ( not a full HTML structure ) to templates, then you might be able to provide some shortcodes for the users to use. The risk of messing with templates here is very low.