It's section 13.004(a) that contains the prohibition on transcription, copying, or otherwise recording a telephone number:
http://www.statutes.legis.state.tx.us/Docs/EL/htm/EL.13.htm
Sec. 13.004. RECORDING AND DISCLOSURE OF CERTAIN INFORMATION BY REGISTRAR. (a) The registrar may not transcribe, copy, or otherwise record a telephone number furnished on a registration application.
(b) The registrar may transcribe, copy, or otherwise record a social security number furnished on a registration application only in maintaining the accuracy of the registration records.
(c) The following information furnished on a registration application is confidential and does not constitute public information for purposes of Chapter 552, Government Code:
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That comes before the other items that are considered confidential. While not under the 'confidential' label, there appears to be an outright bar on copying the numbers. Unfortunately, O'Keefe got this one right, and Battleground Texas may have violated the statute.