ok, well it seems to be more of an electrical problem more than mechanical, so u have fuel pressure but it seems to be starving after running for a little while
best bet, swap the battery with one that yu know is in good condition and see if it makes a difference, that way u will iliminate one cause of problem
your battery may read close to 12v but its not easy to tell if its still ok goin off this reading, u have have dropped a cell in the battery causing carging falure, even if its a new battery thses thing do happen
if its not that then check the ecu, might be that the incoming power(12v) cuts out when running or the ignition lead (12v) cuts out when running, this is the lead that will have power when turned to idnition and will have power when car is running or not but only whe nkey is turned to ignition
if this does not work see a auto electritian
cheers buddy