So, you would create an “update record” form in Fillout that links to your original source base, and the form would contain the attachment field.
Then, back in your original source base, you would create a new formula field that Fillout provides for you. This will give you a unique record link for each record that will take you to your “update record form” in Fillout.
Optionally, you can create a button field that uses this formula.
Then, sync the button field (or the formula field) over to your destination sync table, and your users will have a button that they can click on the edit the attachment field in Fillout.