We did the same thing with a big heavy lump of Canadian cedar. We plastered and painted then decided the rather stark fireplace needed finishing so bought this piece. I measured carefully, then put two large dome-headed screws in the wall with brass mirror plates sunk into the back of the beam. After carefully adjusting the depth of the big screws, I was able to drop the beam onto them where it sits, looking exactly as if it's part of the fireplace. In future if we want to redecorate we'll simply lift off the beam so as to avoid messing around with masking tape.
Doing it this way would avoid the problem of the wood shrinking and leaving a gap around the plaster.
Rather than try to measure to within 1-2 mm, if I did the job now I would simply drill and fit the screws then make a template out of stiff board to show me the required distance between the holes in the mirror plates.
If standard mirror plates weren't strong enough - and they should be as they will be in tension - then you could get some square pieces of steel plate and drill & slot them to accept the two screws. You could screw these plates into recesses in the back of the beam, even setting them in Plastic Padding or Araldite if you wanted to make it really strong.
If your beam is really heavy you could scale this up by buying a couple of long coach bolts and setting them in holes in the chimney breast, cemented in with a two-part resin that's made exactly for this kind of job. However you would lose the depth adjustability so threaded rod with a couple of nuts might be better, almost like mounting a WC or a basin to a wall. Alternatively, could you drill and bolt right through to the inside of the fireplace? You can buy bolts or studs with one end tapered and threaded to screw into the back of your beam.
When replastering the chimney breast, make sure you chip off the old plaster for a good distance away from the fire, otherwise the new plaster will expand and contract with the heat at a different rate from the old and cracks will soon appear along the join. Best to do the entire front face so the stresses are evenly dissipated.