Have you pulled codes from the ECM? Apparently the codes p1340/17748 are associated with timing chain problems. That might help you verify the diagnosis.
Timing chain failures are rarer than belt failures, but they are just as catastrophic. A timing chain failure can cause enough damage to effectively total the car. You definitely want to attend to this, and it is logical to replace the water and vacuum pumps while you are there.
The engine shaking is due to something being out of balance, it seems possible that the timing could be far enough off to unbalance the power production and cause the shaking.
If you are inclined to learn about cars, and have the time to do some work on your car, you might consider doing the chain yourself. It's not a trivial project, according to this post, DIY: 2.5l Timing Chain Replacement, from VW Vortex the VW estimating manual suggests about 14 hours for the task and the timing chain parts (in 2011) ran to about $400 and specialized tools costing $200 were also needed. The author of the post said it took him about a week and half to do the work. From that perspective the estimate from VW doesn't sound too bad. Keep in mind that projects like this (either at a mechanic's workshop or DIY) tend to grow. There will be accessories and seals that make sense to change and you may find that you want to do work on the clutch or renew engine mounts… Be sure to get a good picture of the scope of work that anyone bidding on the project is including in their estimate and also ask about what other items might make sense to replace or inspect at the same time.
Some likely candidates would be:
- Oil seals
- Serpentine belt and tensioners/tensioners
- The water pump
- The clutch if you have a manual
There will also be some fluids (oil, antifreeze).
The catalyst is a whole other question, I don't see it as being related to the shaking problem. An option with the catalyst might be to "delete" it depending on local regulations and how the ECM works.