Depend on badge-maker's public interface instead of relative imports

:clipboard: **Description**

There are 3 features we need to expose in the public interface of badge-maker before we can remove our relative imports and depend on the badge-maker package.

- https://github.com/badges/shields/issues/4947 - logos
- https://github.com/badges/shields/issues/4948 - move JSON rendering from package to server
- https://github.com/badges/shields/issues/4949 - links

This also should help with https://github.com/badges/shields/issues/3370

Start by reviewing issues #4947, #4948, and #4949, which describe the three badge-maker features needed before this change. Confirm how those public interfaces affect the current relative imports; done means Shields depends on the badge-maker package instead of those imports and also addresses the connection to issue #3370.

Written by the indexing model from the issue text.