Should I run iBGP between both routers, in site A and site B.
ISP(AS 3000) is the primary MPLS for all sites (A,B,C&D).
Apologies for not explaining this very well.
THe key thing to understand is that a more specfic route is always chosen over a less specfic route and a default-route is the least specific route you can have.
So lets look at the solution from the perspective of Site A, although exactly the same logic applies to site B.
At the moment you are redistributing BGP into EIGRP on both R1_A and R2_A. So the networks you advertise from Site B via BGP will be received by R1_A and R2_A and redistributed into EIGRP.
So the internal devices within Site A see 2 possible routes to each network within site B ie. via R1_A & R2_A.
Now the idea i suggested was to continue to redistribute the networks received from BGP into EIGRP at site A on R1_A. Lets also assume that you stopped the redistribution on R2_A. With this setup your internal devices in Site A would only see one path to Site B and that would be via R1_A.
In normal conditions this is the path you want traffic to take. But obviously if R1_A fails then there is no other path being advertised to Site B. Now you can't just add redistribution of BGP into EIGRP back on R2_A because then the traffic can take either path and you only want it to take R1_A under normal conditions.
So you don't redistribute the networks in site B into EIGRP on R2_A. What you do is add a default route on R2_B and then redistribute this into EIGRP. This default-route will have the same metric/AD as the other routes redistributed on R1_A so the internal devices within Site A will see the specific network routes redistributed from R1_A and the default route redistributed from R2_A.
And because a more specific route is always chosen over a less specific route the traffic will always go via R1_A and not R2_A. If the R1_A link fails then the internal devices within Site A still have the default route via R2_A to get to Site B. If R1_A link then comes back up the more specific routes will be redistributed into EIGRP again and traffic will take the path via R1_A.
Now you could actually just change the metrics when you redistribute on R2_A but i wasn't sure of the complete topology of your networks in Site A and B and metrics can take a fair bit of tweaking sometimes to make sure traffic is using the right path. The default route solution does not rely on tweaking the metrics and so, in my opinion, is easier to implement.
Hope this explains it in more detail. If you have further questions or i have missed your point then let me know.