Is the check engine light on?
how did it run up to this time?
did you check the valve timing? good compression does not mean its in time
it could be many things but the 3.5 in the LH body had a habit of jumping time due to idler pulley bearings going bad and water pumps locking and releasing when cool, you say that you can see the timing belt, how much of the front timing covers have you removed, may be able to get a ball park with out full tear down, We have had many of these towed in off the pike with similar problems
With the black tin cover part way off, all the bolts in the cover that you can get to easily from the top removed, you can pry the cover back some with out bending it to far about two inch and then look down from the left side with a flash light and see the water pump and pulley, look under the center of the pump pulley for a trail of coolant or melted seal, if ok look down to the crank pulley and then a little more to the right you should be able to see the tensioner pulley, i use a small telescoping mirror to see the face of the pulley to see if the seal has failed, if it looks torn or melted or you can see the ball bearings it has failed,if all looks ok you can turn the motor over by hand from the front crank bolt slowly until the tdc indicator on the crank pulley lines up with the mark on the front cover, then look at the cam pulley the dot on the pulley should be between the two dots on the upper cover ( to see both you will have to pull the other upper cover on the left ),if it is 180 degrees off turn the crank one more full turn and then check, with out removing both covers this is a quick way to see if you are in the ball park, also watch the water pump pulley while turning the motor to see if it turns and does not wobble,only turn motor clockwise, let me know how you make out
If the converter is your problem be sure to check the intake manifold for leaks, these gaskets go bad and are impossible to see, with the car running carefully spray some carb cleaner under the front of the intake manifold, right below where the thermostat is, spray a fair amount trying to get it to go to the back of the intake, spray from both sides and watch for noticeable change in the way it runs or it may just stall out if they are leaking,if you find a change you should replace the lower intake gaskets, it is not comon for the catalytic converter to go bad on its own, I don't think that I have changed more then two of them in the shop
Let me know how you make out and if you need more information
Did you replace the converter?
Did you replace the pcv hose and the valve also, be sure to check the whole pcv system to be sure that it breaths good
How does it run now?
Converters normally don't go bad on the LH, it looks like this is the cars third one, something is causing them to fail, oil consumption? fuel drip from leaking injectors? vacuum leak? loss of coolant with out seeing any drip's / may be burning it, Fuel additives containing lead, I would be sure to give everything a good once over to avoid the cost of having to do it again at a latter date, how old is the tune up misfires can melt the converter, Also check the EGR system the tube from the valve to the rear of the intake may be plugged with carbon, no EGR flow makes high combustion temp's and high NOX emissions which can also melt the converter
Let me know how you make out
Well with it gutted you wont have to worry about it, but it is against federal emission laws to do so, so with that said.... as long as you have given the whole engine management system the once over and all is good then it appears that you are all set, there really is nothing other then what I already listed that I can think of that could be causing repeat failures, I checked and saw no recalls for anything related to this or service bulletins
Let me know if you need anything else
The best comparison I can think of is when building a high performance motor
When you install the camshaft you can put it in on the factory suggested marks or you can degree the cam shaft, looking for the highest compression that you can reach, so with a dial indicator, etc, etc you put the cam in the engine and find its center line and the you turn it advance/retard to find the best combination for the most compression which can be 2/4/6/8,etc degrees away from where the factory says to install the cam, now on a carburetor car with a distributor this change in valve timing can be compensated for, but on a car without a distributor and with fuel injection if the belt jumps, (on the 3.5 this normally happens due to the water pump locking and unlocking, and the belt skips two or three teeth) the compression can climb due to cams "degree-ing" them selfs but the computer is firing the coil and injectors at the wrong time, so when cranking it sounds good and almost will try to start but wont,
Chrysler has specs for the 3.5 at 150 psi and not to vary more then 25% from cylinder to cylinder and not to be below 100 psi, from working on many of these I commonly see from 150 to 165 when all is good, normally when it does not run and i see in the 200 range the belt has skipped,but in your case this was most likely due to the exhaust not having any place to go, I have just seen lots of belt problems over the years and almost no catalytic converter problems and have been bit before by not looking at the t-belt early on