I'm with delcowizzid and VrWagz1; it sounds most like a BCM (Body Control Module)/security system issue.
Unless the LPG system is a factory one, it won't be disabled by the BCM. (A factory system can be ID'd by looking for a ~30X30X15mm box attached to the lock off solenoid under the inspection cover on the LPG tank).
Because the car runs on LPG and starts with fuel added into the intake manifold, the ignition system is OK. The CAS sensor and the DFI (Direct Fire Ignition) module must be working. (No CAS = no fuel injector pulse and no spark, and no run on LPG or the added fuel; no DFI = no spark and no run on LPG or the added fuel).
The theft deterrent system (in the BCM) prevents the PCM (Powertrain Control Module) from working, and therefore there will be no injector pulse, but the ignition system operates separately until the PCM provides its input.
There will still be spark through the DFI, fixed at 10 degrees BTDC (Before Top Dead Centre, TDC = with the piston at the top of the cylinder on the compression stroke), but no fuel added or ignition timing control without the PCM.
To check for injector pulse, pull one or more of the injector plugs off and with ignition on check that there is power to one of the two wires with a test light or voltmeter. Then check that the power wire is earthed through the other wire when the engine is cranked over. Only use a test light that draws less than 300mA when connected to 12.6V.
The fuel pump will operate for a couple of seconds when the ignition is first switched on but not after that without the PCM active, so that sounds OK.
If you have another spare key and remote, try that as a check of the key/remote operation.
The contacts around the ignition switch may not be connecting properly with the key.
If it's not the BCM, maybe look at the fuel changeover switch.
Check for fault codes too - see the 'How To' section.