If you haven't reviewed the steps on how to set a form as invisible, please refer to the Add A Form article before reading below.
If you defined an "invisible" form that you need to assign a pet, head over to the pet profile in question, and click into their Forms section:
Then select the "Attach form" button:
Finally, attach the form that wasn't visible to the client to fill out themselves:
In the example above, the pet is a chicken (versus the typical dog or cat), and the pet sitter needs the client to fill out a form that they defined. As soon as the pet sitter selects the form, it shows up in the profile of the pet and visible for your pet owner to be able to fill out the extra details.