Tough as there is not much evidence for late 12th early 13th, but we do have cloth records for purchase of cloth in a single colours, particularly by the second barons war, and the earliest field badges apart from a cross.
you are on the right path though
Lord in heraldic surcoat, shield and horse barding, ony he or his sons would wear this, not the whole retinue.
Gonfanon bearer, A knight in his own heraldry or a sergeant in tabard block colour, with a badge left, centre or right breast.
retinue block colour or perhaps parti coloured tabard if they wore one, most often just in a gamberson either natural or in a simple colour to achieve light greens, browns, pinks and mauve with a field sign sewn on the chest or worn on a arm (armband) in a block colour.
The only reference I have for Edwards I livery was red and white with no reference to a cross, whilst artwork has a white tabard with a red cross, but it is 14th century.
cheers
Matt