Mike - just a thought: would the leathers from a 'bulldozer' type water pump fit the tyre pump?

I don't know if that's what you call them over there . . . they're a double acting reciprocating water pump that used leather washers as seals.
They were also common in windmill pumps.

These washers are still readily available here in N Z in sizes ranging from 1" to about 6".

( They're also pigs of things to fit . . . the second one goes in easily, but the first one . . . Oh boy!)