Really, looking at the country of manufacture (COM) stamp is more telling. Using that bulldog for example, the 'football shaped' MADE (arched over) --IN --ENGLAND (arched under) is consistent with Comoy's as the maker as it is one of the stamps they were known to use in the 30's through about 1945.
The LIVERPOOL brand is not one I have heard of before, but Comoy's downstreamed into more sub-brands than any other manufacturer (literally dozens).
The shape number with 'seconds' sometimes remained unchanged between tiers, but not always, so you can't always rely on that. The LIVERPOOL bulldog does have a known Comoy's number.
Doc