If Mahoney didn't authorize construction of this chapel, then he's building it extra-ecclesia, that is, without proper authority.
Who will bless it and consecrate the altar? Isn't that normally the function of the bishop?
Yes. If he intends to use it for the SSPX, then an SSPX bishop (Williamson or Fellay) will have to consecrate the building.
If I recall correctly, a church must be authorized and consecrated by the diocesan bishop, whereas a chapel need not be. Does that sound right to you?