Black dwarf binary systems
I see two real questions here. First, whether it's possible to have a black dwarf with a companion object. For a given black dwarf, this is unlikely, since the orbits would likely be unstable at the time scale required to produce a black dwarf. Given the size of the universe, however, it's not out of the question. A black dwarf could even capture a companion star after cooling down, although this would be extremely rare.
Assuming it's possible, the next question would be whether the supernova produced would be the same as a normal type Ia supernova. The supernova is generally triggered based on the critical (Chandrasekhar) mass which triggers carbon fusion. Since it's dependent on mass, it would likely be a &quot;normal&quot; type 1A supernova. Keep in mind not all 1A supernovae are identical to begin with, although they can still be &quot;normalized&quot; to function as a fairly reliable standard candle.
One thing to note is, the accretion process would cause the black dwarf to heat up, meaning it would not remain black.