I think the problem you face is that the JCE editor integrates its file upload feature, so it is not easy to isolate that feature into a stand-alone Joomla type field. Subforms do allow you to add a 'file' type field - a type that allows you to select a file and upload it. (I have built a subform with this field type as a way for the client to add and reorder images.) The challenge with the 'File' field type is that upon submission, it is your responsibility to supply the code that will copy the uploaded file to the directory where you want. Also note, I pair the 'file' type field with a 'media' type field so that together they serve as a single field - as the first is used to declare which file to upload and the second is used to record which file/image is currently loaded for this field pair.
Of course, a subform allows you to add an editor to each set of fields - and the field declaration can set the attributes to request JCE as the preferred editor.
If you are concerned about the form having its own layout that suits your intended purpose, be assured that you can create your own template for how each subform record view is laid out - you do not have to use one of the Joomla-supplied subform layouts.
In short, I think you can accomplish what you seek -- as long as you handle file uploads with the Joomla field type for such, instead of trying to use JCE's file upload feature.