launch custom profile from Edit button?
You can do it with a link but I'm not sure how to get the link to display somewhere without some customization. For example if the contact id is 2 and the profile id is 3 then this url will open the profile to edit the user:

https://your-website/civicrm/profile/edit?cid=2&amp;gid=3&amp;reset=1

The profile id is what you see in the url when you're on the admin page where you created the profile, and you hover over the Fields or Settings link, e.g. in this example it's the gid=3 part:
https://your-website/civicrm/admin/uf/group/field?reset=1&amp;action=browse&amp;gid=3

EDIT: There is a way to get to the desired page just using the UI but it's a couple steps:


Go Search - Find Contacts. Pick some criteria that will find your
contact.
In the results list check the checkbox on the left next to
the contact.
From the actions dropdown at the top of the results list, choose Update Multiple Contacts.
It will then redirect to a page where you pick the profile.
Then on the next page it will present a version of the edit form.