Hey Eric,
You mentioned specifically if there is a method using CSS, however I can offer an out of the box method for you.
The idea is to use a Checkbox field (for example we could name it "Hide Field"), and set a Field Rule up so when this Checkbox field is Checked, we will hide the field. When the Form is submitted, the field will not appear on the Form.
This offers users the option to hide the field before submitting the Form and having it Save to Laserfiche.
If you are looking for a fully automated solution, you could look at doing something very similar, but having Javascript check the box for us when the necessary conditions are met (for example when all necessary fields are filled). If you can get this to work, then you can also look at hiding the Checkbox field so that this method is automated only, by configuring a Field Rule to Show this Checkbox under an impossible condition (my preference is having a Form Submitter field Read-Only and filled by default, which will never be empty).
As a final point, all field values are stored as variables within the Forms instance details, no matter if the value is hidden or not.
I hope that this has been helpful!