Unless someone has created a plug in that offers that capabiiity, I think you'll need to go with the overlayed fields.
Some years ago, I was working on a project the objective of which was to import into FMP (9 or 10, I believe) data from pdf forms. I discovered that the pdf form has a relationship with Excel, to output data from the form (and a number of forms at one time) into an Excel sheet. The data could then be imported into FMP from Excel normally, field names mapped to or the same as the fields in the pdf.
I also found an online software company at the time that sold a plugin ($75, I think, for one seat at the time) that eliminated the need for the Excel step.
Reversing the order of export / import might still work, with Excel, or maybe with the plugin.
It's been awhile since I did this, and I'm not sure what the final objective is for you, but this is what I remember.