You must do DFT calculations of the electron structure. Then you must be able to calculate superexchange overlapp integral. This is quite difficult. You make take the help of someone who knows ab initio electron structure calculations. The other simpler method would be to determine the spin wave dispersion of your crystal by inelastic neutron scattering. For this you need large single crystals. Then you have to model the spin wave dispersion with a Hamiltonian (Heisenberg for example) that will contain all the exchange terms you mention in your question. Then you must fit the experimental spin wave dispersion by varying exchange interactions and then you find the exchange interactions from the fit.

You can contact Tanusri Saha Dasgupta of SNBNCBS, Kolkata and request her to help you. She and her does such ab-initio calculations. If you can mange to grow large single crystals then contact then you should write up a ILL research proposal for measuring spin wave dispersion.

Thank you sir for your valuable and important suggestions. 

A loud talk: whether respected Dr. Tapan Chatterji will approve it?

It can lead to very interesting results as:

[A] δA1, δA2, δB1, δB2,

The sign of J is determined by magnetic moment signs( normally, sign is avoided)

JAA= δA1-δA2

JBB= δB1-δB2

JAB= δA1- δB2

[B]One can calculate KAB

KAB = 4π2. j AB / h .γ A .γ B -------------------------- [1]

γA and γB are reported in the literature.

[C] After calculating KAB, one can calculate ONE OF MOST IMPORTANT PARAMETER called SPIN HAMILTONIAN[H Spin ]

H Spin= KAB. mA. mB

You have values of mA. mB ( magnetic moments of A andB) with you.

[D] This value of the Hamiltonian can be confirmed by:

H Spin = 6.023 j AB. IA. IB.1017MHz mole-1 .

I don't know which particular compound you are interested in, but maybe this article would be useful for you:

http://journals.aps.org/prb/abstract/10.1103/PhysRevB.19.499

Thank you Mr. Manohar Sehgal and  Marjana Lezaic for your valuable suggestions.