Christine,
Very difficult to be precise as you know but it is towards the earlier backactions. One can tell by the style of the 'action bars' which I describe a 'droopy' or 'wrap round', the button safety rather than a beetle back and the 'ball' rather than 'Webley' fences. I estimate the 'later' styles started between numbers 5600-5700. I have calculated (very roughly) that Blanch produced about 50 numbered guns per year.
6063 is the first gun number I have as proofed after 1896 and I have dated 5450 to about 1884 so that puts 5578 at approx 1887-8.
Hope this helps,
Toby