From Lessons In Tanya, Iggeres Hateshuvah, ch. 7:
There are two distinct states of Divine compassion, indicated by the
terms “Merciful Father” and “Father of Mercy”. The former term (אב
הרחמן) merely signifies that G‑d possesses the attribute, or middah,
of mercy — and since middah means not only “attribute” but also
“measure”, it refers to a finite quality of mercy. The latter term
(אב הרחמים) stresses the fact that G‑d is the father, or fountainhead,
of all mercy. Arousing His essential quality of mercy “from the Source
of mercy” thus means arousing His infinite measure of compassion —
- and thus we use the phrasing אב הרחמים at times when this essential quality is more evident (for example, at Minchah on Shabbos, which Kabbalistically is a time of great Divine favor, רעוא דכל רעוין).
As msh210 pointed out, though, I've misunderstood his question: it was about why אב הרחמים has a kamatz in Chabad siddurim, not about the difference of אב הרחמים vs. אב הרחמן.
It seems that the Lubavitcher Rebbe zt"l himself had the same question. In his notes on the Siddur Tehillas Hashem ("Rostov Siddur") of 1941, he comments that it would seem that the correct grammatical form should be with a patach. In a later series of notes on Siddur Torah Ohr, he makes the same observation but concludes, evidently after consultation with his father-in-law and predecessor R. Yosef Yitzchak Schneersohn, that it should be left with a kamatz.
No reason is given, although perhaps it is indeed related to the idea above, that אב הרחמים represents the idea that the אב Himself is רחמים rather than just possessing that quality.