template_project_id is unknown or invalid when trying to create project from template via API

Summary

When using the API and trying to create a new project from a template, an error is thrown suggesting the template project does not exist when actually it does.

Steps to reproduce

curl --request POST \
  --url 'https://gitlab.com/api/v4/projects/?name=test-project-api&namespace_id=10226105&use_custom_template=true&group_with_project_templates_id=10226098&template_project_id=22702533' \
  --header 'Authorization: Bearer TOKEN'

Example Project

https://gitlab.com/181657

What is the current bug behavior?

An error is thrown that the template project is unknown or invalid.

What is the expected correct behavior?

The project should be created based off of the template project.

Relevant logs and/or screenshots

Output of checks

This bug happens on GitLab.com: 13.7.0-pre 7127ddad523

Workaround

See #290003 (comment 1366916702)