Question about LDOs and the practicality of this application
As I read the datasheet, the TPL7407LDR requires at least 8.5 volts on the COM pin - higher voltages are fine.

I would suggest a 9 volt switching regulator (or 12V->9V DC-DC converter) rather than a linear LDO.  With an LDO, you will be dissipating about 12 watts in the 9 volt regulator if you start with a 12 volt power supply.