Gareth,

You are correct on the origin of the mitre cap, but the mitre *became* the bearskin cap over the course of the 18th century, however, up until Waterloo, it was restricted to grenadiers only (The battalion was divided into eight companies, one grenadier, one light and six "battalion" or "hat" companies). On some occasions, all the grenadier companies of a brigade could be formed as an ad hoc composite grenadier battalion (likewise, composite light battalions could be formed before the advent of specilist Light Infantry).
After Waterloo, the headdress concession to the Foot Guards was that THE WHOLE BATTALION (not just the Grenadiers) would wear the grenadier bearskin.

Regards

Walrus