They are 31 weeks old and only 16 are laying and this week only 12.
How are you figuring how many birds are laying? The number of eggs you collect will not give an accurate count as all birds don't lay every day.
Free range birds may have found a 'better' place to lay.
Free range birds sometimes need to be 'trained'(or re-trained) to lay in the coop nests, especially new layers. Leaving them locked in the coop for a week or so can help 'home' them to lay in the coop nests. Fake eggs/golf balls in the nests can help 'show' them were to lay. They can be confined to coop and maybe run 24/7 for a few days to a week, provided you have adequate space and ventilation, or confine them at least until mid to late afternoon. You help them create a new habit and they will usually stick with it. ..at least for a good while, then repeat as necessary.
2 small with four in each (5 capacity) and 9 in the large coop (10 capacity)
How big are these capacities in feet by feet?
Dimensions, and pics, could help immensely here.
Should we band them and make sure the same ones go into the three different coops or does that matter?
Banding them won't
make them go into the 'right' coops, but it may help you see who is going where. The problem with multiple coops is that too many may try to go into a too small coop....which could cause stress which could affect laying.
Keeping them confined to the runs attached to the coops(I think each coop has a spearate run?) would keep them sorted.
How old are your birds?
...and...
Where in this world are you located?
Climate, and time of year, is almost always a factor.
Please add your general geographical location to your profile.
It's easy to do, (laptop version shown), then it's always there!