I'm not sure if this will answer your question as you are stating that you want to render a region in the menu template. You can't do this in the traditional sense but you can use Twig Tweak to render blocks pretty much in any template assuming you have the right context.
See their help pages for good documentation on how to do this.
As an example, to render a block in any template, you'd write something like this:
{{ drupal_block('plugin_id', {label: 'Example'|t, some_setting: 'example', setting_array: {value: value}}) }}
You can also render views and full entities which comes in handy as well, for example:
{{ drupal_entity('block_content', 1) }}
In general, Twig tweak is a super handy module that we use pretty much on most D8/D9 projects.