From your description, I assume that you are reader-extending with Acrobat Pro, saving the rader-extended form, and then opening the form in Reader.
If this is the case, you only have some of the possible reader extension rights. The others are available through LiveCycle.
The list of all possible rights are:
1.OID_1. Form Fill-in
2.OID_2. Form Import/Export
3.OID_3. Form Add/Delete
4.OID_4. Form Submit Stand-Alone
5.OID_5. Spawn Template
6.OID_6. Signature Modify
7.OID_7. Annots AddDelModCopy
8.OID_8. Annot Import/Export
9.OID_9. BarcodePlaintext
10.OID_10. AnnotsOnline
11.OID_11. FormsOnline
12.OID_12. EFModify
You can probably deduce the use of the rights from the name.
Acrobat Pro (8 and 9) enables 1,3,4,6,7,8. The right that you need for exporting data is 2.
Sorry... the use case you describe is outside the bounds of the expected use of Acrobat users. In other words, it is by design; you should move up to LiveCycle and then start building processes and backend functionality around this as well.