In a FIDE-like game, I would expect the Immortal to be much weaker than the
Mamra, which does not require support to pass through a threatened square
(as long as the threat does not come from a pawn) and can easily checkmate
the enemy King completely unaided (and regardless of any non-pawn
defenders). The page you link advocates sacrificing a Queen and a Rook to
create a hole in the opponent's pawn wall through which the Mamra can
charge, which suggests the Mamra is worth significantly more than a Queen
(and that wouldn't surprise me in the least). The Immortal poses no
remotely comparable threat that I can see.
However, the Mamra's value probably varies wildly depending on the other
pieces on the board, both because it is a highly specialized piece and
because it is vulnerable only to a specific type of enemy piece. The
Immortal's value also probably varies more than most, but not to the same
extent.
Relying *entirely* on testing to balance a piece is 'brute force' in the
sense that it makes no attempt to leverage information unique to the piece
being tested, and is not even CLOSE to the speed or accuracy you suggest.
What you describe, where you 'estimate' (by unspecified means) a value
that is somehow magically within +/- 1.0 pawns initially and even more
magically within +/- 0.5 pawns the next game is not balancing based on
testing, it's balancing based on intuition (with exceedingly optimistic
estimates of accuracy). Intuition occasionlly works very well but often
fails catastrophically and is completely irreproducible. And yes, that is
no doubt the primary means by which most CVs are balanced, but I was hoping
for something a little more insightful.