Absolutely. It won't lay as well as a pure RIR or hatchery stock production red but egg production is indeed hereditary and is contributed from both parents.
If you wanted to you could hold over that cock and breed him over his offspring you'd again raise the production in the line. I don't know how available hatchery or higher production birds are in the Bahamas but if scarce this would be a great way to obtaining a more useful backyard flock.
To really ramp up production just pay close attention to this crosses offspring and tag the earliest pullets that come to lay and those that you know lay the most. Only use the best and earliest producing pullets for mating back to the sire.
Ok, that sounds good. It would be like line breeding correct? Yes I have a RIR rooster that will stay around for years to come, that I imported as an egg and hatched last year. Thanks.