I am using the Better Beamer not the Magmod, but I believe they are similar products.
Where I live, we have hills all around us and have plenty of big trees, so the birds are very rarely in sunlight, they are ussually in heavy shade.
Without the help from flash I struggle to get any useable shots as I need to go to iso3200 and above. I use the D7100, Tamron 150-600 G1, handheld.
After much trial and error, I have found what works for me is to go fully manual, set the shutter speed on 1/200 or 1/250, aperture where your lens is the sharpest ( mine is from f8 - f11 ) then use iso to expose for the background ( sometimes the background will have to be a bit dark because you obviously dont want the iso going too high ) then use the flash power settings to get the right amount of fill light on the bird, this can take a bit of trial and error. I ussually start at 1/8th and see what it looks like ( my usual range is anywhere from 1/16 to 1/2 ) When you start getting up to 1/2 or full power it starts to look unnatural.
These are a couple I took the other day of a Kookaburra in my backyard. They were either 1/8 or 1/16 flash power. The extra light from the flash really helps to bring out the detail on the bird that I just cant get without some fill light help.
EDIT- I just noticed in the exif, the exp comp says -.33, because I was in manual this wount come into effect. I just forgot to reset to 0 from when I was on I was on A priority or auto iso the last time.