Very nice. Could I have the frame number please? It is likely on the right side of the seat lug.
Re engine: depends a bit on the frame number. The early version of this frame was usually fitted with the overhead-inlet-valve Fafnir single, thus the very narrow lugs. When Healing gained access to Precision engines, they fitted 500s to this frame using aluminium spaces between the frame lug and the engine plates. Some of Healing's customers - maybe even Healing themselves - fitted the 4 1/4 hp Precision (600cc) to this frame, but to do so the front down tube had to be bent forwards because of the bigger crankcase. It would seem wrong to do this now. If a 600 Precision were fitted to this frame it would have had Precision on the magneto chain case. PRECISION BIG FOUR was a Healing trade name that was only used on the "new" bikes (in new frames) that came out around August 1913.
Anyway, I can guess engine based on frame number, as Fafnir engines in this frame stopped when the new Peerless frame came out in August 1912, but Precisions continued. It's such a nice frame - many have been cut down to lower the seat - that it would be a pity to put the wrong engine in it. Missing petrol tank is Davison and hubs were Chater Lea.
Leon