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I let the computer run games all night (at 40 moves/min), and the first
results indicate that the game is biased in favor of the Spartans: they
score some 70-75%. For comparison: in FIDE Chess Pawn odds results in a 68%
score, so it seems the advantage is slightly more than a full Pawn.

Noticeable is the low draw rate: only around 15%. (In FIDE this is ~32%.) I
guess the end-game value of the Hoplite is larger than that of a FIDE Pawn,
making it much more difficult for the Persians to draw games where you are
a Pawn (Hoplite) down. For instance, KHK has far fewer draw positions than
KPK. (On a cylinder board it would have none at all; you are dependent on
the Hoplite running into an edge in order to stop it, and even unprotected
Hoplites cannot be approached to capture them, and even can advance through
zugzwang.)