What is not described on that page is that, if you want your form to save these field values, you can add the argument "persist": 1 to the JSON options for each field.
You can control those fields, and have them in a Form, but complex fields, let's say an ACF gallery, for example, will not display and be editable as such in Forms. It will assume a simple input and likely break your complex field.
So usually you would need to work with possibly simple fields if you plan to control them with Forms.
Sometimes, it needs a bit of trial, to see what fields you can control and which not, as each field of each plugin may be different.
Is there a way to register this post type and custom fields so that Toolset recognizes it?
You can as well control Custom Fields with Types, not only with Toolset forms.
There, the same applies if the fields are complex, they will likely not work the same if controlled with Types.