Some thoughts....
Typically one does not want to store pieces of tightly interrelated information in different systems. The chances of things getting out of sync is significant and now instead of one problem on your hands you have two. One thing you can do with Mongo though is use it to pipeline your data in or data out. My preference is to keep everything in PostgreSQL to the extent this is possible. However, I would note that doing so really requires expert knowledge of PostgreSQL programming and is not for shops unwilling to dedicate to using advanced features. I see a somewhat different set of options than you do. Since my preference is not something I see listed I will give it to you.
You can probably separate your metadata into common data, data required for classes, and document data. In this regard you would have a general catalog table with the basic common information plus one table per class. In this table you would have an hstore, json, or xml field which would store the rest of the data along with columns where you are storing data that must be constrained significantly. This would reduce what you need to put in these tables per class, but would allow you to leverage constraints however you like. The three options have different issues and are worth considering separately:
hstore is relatively limited but also used by a lot of people. It isn't extremely new but it only is a key/value store, and is incapable of nested data structures, unlike json and xml.
json is quite new and doesn't really do a lot right now. This doesn't mean you can't do a lot with it, but you aren't going to do a lot out of the box. If you do you can expect to do a significant amount of programming, probably in plv8js or, if you want to stick with older environments, plperlu or plpython. json is better supported in 9.3 though at least in current development snapshots, so when that version is released things will get better.
xml is the best supported of the three, with the most features, and the longest support history. Then again, it is XML.....
However if you do decide to go with Mongo and PostgreSQL together, note that PostgreSQL supports 2 phase commit meaning you can run the write operations, then issue PREPARE TRANSACTION and if this succeeds do your atomic writes in Mongo. If that succeeds you can then COMMIT in PostgreSQL.
Best Answer
You can look into tailing the oplogs of the shards yourself, thereby sending the data to a cluster in a remote data center incrementally. Similarly you can look into tools like mongoriver from Stripe or mongo-connector from MongoDB labs if you don't want to do all the work yourself from scratch (but then are relying on a third party keeping that tool up to date or an unsupported labs project). You will still need to wrap these utilities in some logic based on your needs and do some manual data verification to validate date between the two sites, and getting point-in-time snapshots of a cluster is non-trivial.
The only "out of the box", supported, way to do this (that I am aware of) is to use the on-premise backup capabilities of Ops Manager which is a paid offering from MongoDB themselves (full disclosure: I used to work for MongoDB). It's a complicated process to keep a full copy of a sharded cluster in a separate location, so really your choices are to invest your own time and resources, or pay for a product to take care of it for you.