hi there,,,dont know why,,but as i read your post i had a hunch that Frederick was in the military,,,and i think i was proved right

There is a birth recorded in the regimental birth index of a Frederick Pearson
the regiment of the father was the 22nd foot..he was born in 1835 in Lambenhurst Kent volume 1915 page 75
with this reference you can order his birth record from the GRO.
The reason you cannot find Frederick in some of the censuses is that he himself attested for the 22nd foot in INDIA in 1850 age 15.
he was number 3066
he served in India for nearly 5 years,,,Malta for nearly 6 years and New Brunswick for 3 years,,,he was discharged in 1874 and his intended place of residence was Chester,
you have music in your forum name,and interestingly he was a Musician by trade.