Before getting onto anything else like the CPS. Have you thoroughly checked the primary ignition components on your car to ensure they are all new, or very recently replaced? I may have missed it, but I've not seen anything to suggest this has been done?
By primary ignition components I mean the spark plugs, the leads, the distributor cap and rotor, the coils, the coil trigger modules, and the underbonnet wiring for the coils? The Marelli ignition V12s have what are effectively two ignition systems and all sorts of rough running maladies are caused when one or both starts to play up (including the infamous 'Marelli fire'). When the engine is hot and resistance of some components increase, the problems are more noticeable. When cold the problems often disappear for a while. Make absolutely sure all the primary ignition components are either new or very near new before trying to diagnose other problems.
There are two fuel pumps for the V12 and a module that controls them. If the car has been sitting for a long time the second pump (which only operates under 'full load' conditions) will often seize because it doesn't get much use anyway. The control module can also fail, but this is rare. If your mechanic is going to take the tank out to clean it, get them to check both fuel pumps for operation as well.
The V12 has two, almost identical, crankshaft reference sensors - one is crank angle and one is crankshaft speed. One reads off a three pointed 'star' bolted to the back of the crankshaft pulley, the other reads off the torque converter drive plate. They were interchangeable on the 5.3 with Marelli ignition, for some reason on the 6.0 they are slightly different (plug, cable length, whatever). That said, Naki is spot on when he says these rarely fail on the V12.
Like many cars, the crankshaft pulley on the V12 is a harmonic balancer. There is an inner and outer part of the pulley which has a bonded rubber section between the two parts. Over many years the outer section of the pulley can slip in relation to the part bolted to the crank, and since the 'star' is bolted to the outer section of the pulley the two reference sensors are then 'out of synch'. This also causes ignition problems.
The pulley can be checked by either doing a top dead centre (TDC) alignment using the number one piston (PITA), or by setting the engine to TDC using the engine speed sensor teeth on the drive plate (slightly easier PITA). Once the engine is at TDC, if one tip of the star on the pulley isn't in alignment with the crank angle sensor - the outer section of the pulley has slipped.
As mentioned at the start though. Make sure your primary ignition components are all absolutely spot on before doing anything else.
'94 XJ12 (and other toys)