There isn't currently a way to combine templates with inserts from other dialogs like Image Manager and File Manager (a project I want to start sometime soon), but you can use Template Items to create a list of items to select, which will show a dialog for the user to select a file, and add other values like a title.

See Template Items here - https://www.joomlacontenteditor.net/support/documentation/template-manager/template-manager-configuration

So for example you could create a Template Item with the following HTML:

<div class="foo">
    <a href="https://www.joomlacontenteditor.net/${URL:file}" title="${Description}" class="bar">${Name}</a>
</div>

Which will give you a dialog like this:

Please try JCE Pro 2.9.82 RC - https://www.joomlacontenteditor.net/downloads/editor/pro/development

Clear your browser cache after install.

This adds support for size, date, filename and extension variables, eg:

<div class="foo">
    <a href="https://www.joomlacontenteditor.net/${URL:file}" title="${Description}" class="bar">${FileName}</a>
    <span class="date">${Date:disabled}</span>
    <span class="size">${Size:disabled}</span>
</div>

FYI - using :disabled next to the variable eg: ${Size:disabled} will disable the input field, so the user cannot change the value.

The new variables are Size, Date, Extension, Filename (the file name without the extension)

JCE Pro 2.9.82 RC3 - https://www.joomlacontenteditor.net/downloads/editor/pro/development - adds a hidden type for variables, allowing you to use the value in the template but not expose it for editing, for example, in the download attribute the filename and extension values are used but are not visible in the form:

<div class="foo">
    <a href="https://www.joomlacontenteditor.net/${URL:file}" title="${Description}" download="${filename:hidden}.${extension:hidden}" class="bar">${FileName}</a>
    <span class="date">${Date:disabled}</span>
    <span class="size">${Size:disabled}</span>
</div>