If you are ok with a computation of the self-force to linear order, it is best to use linear perturbations of the AdS background. I do not know whether this has been done for AdS in any non-horrible way (it may be non-obviously buried in some paper), but I certainly liked the treatment of linearized waves on dS by Bini, Cappozziello and Esposito.
The obvious problem with AdS are the boundary conditions at conformal infinity, but if you care about some immediate self-force, i.e. not something such as a particle accelerating throughout the whole cosmic history, this should be possible to hand-waive away (but maybe you are even interested in the interaction with boundary conditions because of AdS/CFT, those are unknown waters for me). I don't think AdS3 will be any simpler, but one should expect the solutions to be very different in every such low dimension.