Normally I would have thought that 400W was about on the low limit to justify the extra cost.

Don't get too excited, it will be rare that you see 200W from a 200W panel. As Ghurd said the battery voltage will be above 12v by a fair bit and this knocks the current down so your 16A is most likely going to be nearer 12A.

The final stage of the charging process can not be rushed, batteries just won't accept it.

Full sun with MPPT will give you around 193watts / 14volts = 13.7 amps Not as you calculated. (assuming everything works right, stated right etc.) Several times improvement over direct connection to battery, well worth doing. Else can the panel be rewired for 12v output (or cut in half) would help, still not achieving MPPT potential .

Adding a tracker can be very beneficial, several nice designs out there DIY not costly.

!!!Important discussion on Large batteries!!!!

Solar power is expensive so are LARGE batteries.
Lead Acid batteries are 91% efficient charging when state of charge (SOC) is below 80%.
SOC Above 80% efficiency drops to zero as 100% SOC is reached. Battery is fully charged.
By 90% SOC charge efficiency maybe less than 50%.
By 95% SOC charge efficiency maybe less than 25%.
If you expect to get 5 days only discharging 20%, then your working in the 90% SOC most of the time.
Several factors affect charging efficiency, SOC, battery temperature!, Battery voltage and charger.
The above numbers can be improved... another discussion. (not by a huge amount)

Lead acid batteries are made to be used, using only the upper few percent the batteries may last for many many years, you also spent a LOT of money for them. To keep the large battery charged you will need a solar panel 2-4x larger! And the battery is 4-5x larger than needed.
Over all not a good deal to over size the battery.
Charging a Lead acid battery fully is also stressful, getting above 95% requires a finishing charge raising the temperature stressing the battery!

If you NEED the battery to last forever because you believe the world is going to end (or some other like reason) they by all means over size everything with the understanding in cost today. (Storing a dry battery can last many years... (my choice)).

Reducing the battery so by the 5th day ends SOC=20%, will stress the battery, if occurs frequently like a couple dozen times a year consider using a larger battery. If not then I would not size it any bigger.

Just a thought, by starting out with the smaller battery, save the money you would have spent, you have enough to buy 4 more sets! by the time all are warren out, I bet it will be longer than the time the over sized battery set would have lasted. And you saved 2-4x on the solar panels or generator run time.

Many people with over sized batteries must run a generator frequently to keep it charged, solar panel array is not large enough to keep their loads and battery fully charged. They do not understand why this is happening. Now You know.

Hope this helped!

The 20~25% gain would be about max.  I would figure closer to 10% average.

A couple things will work in your favor.
You won't go 5 days with "No Sun".  It may not be bright, but you will get some charging everyday.
MPPT will put more amps into a lower battery than a fuller (is that a word?) battery, given the same 'sun'.

The 650AH seems reasonable to me.  And that is 3 pairs of 6V golf cart batteries from Sam's Club / Costco, though I would probably go with 4 pairs.
Might look into 2 pairs of L16 or L16-HC if the wallet can handle it.  Interstate is less expensive than Trojan.
It will make a lot more power in summer than the 3 hour insolation.

Size wire to keep losses below 2%.
Calculator bottom of the page:
http://www.powerstream.com/Wire_Size.htm

The longer the run the thicker the wire.
Solar panels are expensive, power loss in the wire becomes expensive.
Yes, AL can be used. Make sure you have the right termination for it, etc.

My last comment about batteries!-)
Buy the cheapest you can, see how long you can make last. (proper care)
If you have not taken care of batteries before, you do not want to experiment on expensive batteries, could be costly. More than one person has ruined a good battery with poor maintenance.
Strongly consider the inexpensive automatic watering systems. Last look cost $6/cell (with shopping).
My golf cart batteries cost $57 each, have 8. three years and counting.
Many claim to get more than 7 years from cheap golf cart batteries.
Better batteries may last longer, yet need 14 years due to cost ( ratio).
I used cost/watts to compare batteries (watts stored = aHr x volts)

12v x 83ah = 1kwhr per day, not much power!

Consider bus bar vs cables.