Just to go over things:
1891 has no Britannia Schooling but three girls named Rose Schooling. Two were born c1880 and one c1894.
The one c1894 is probably the one LKL found and now crossed out.
Of the two born c1880, one is with the parents Ricky suggested and this is the one I think could actually be your Britannia.
I believe that unfortunately the only way forward on this is to start doling out the cash and to send for Britannia's birth certificate ( ref above).
Will you let us know if you do that and what the results are. I will carry on looking for Britannia in 1891 and 1901 but she is being very elusive.
Jen