Bundle fields for entity without type config entity not created
This is not answer to the problem I had but a workable solution nonetheless.
In the end I have created custom entity per each "bundle" that I have initially planned.

It actually resulted in less tables and also it provided option to handle logic that I have initially planned to do via plugins.

The only drawback of this solution are some route handlers where I have to provide the entity type via option parameter in order to be able to use single handler to get a page title, for example, due to the variable name "restriction" in route handlers.

Also in entity references or entity loader I have to use entity type only instead of entity type and bundle but that's totally fine.

I am using single entity base class, the same forms and so on so not that big of a deal(for now).