Is my only option to get a 4-hour train to Berwick?
Despite what the commercial sites infer in their advertising bumf, there is still a lot which is not (and may never be) online.
You could also email Northumberland Archives and ask them for a quote (given the details you have) to research, find and copy the records you require. Depending on how many children they had, how accurate the info you can give is, it 'may' work out cheaper than the train fare/time spent.
(I too am a long way from Woodhorn, I have had many, many copy records from them over the years, and have always found that their charges are not excessive in relation to the time they have to spend to identify the records I require and send me copies)
Boo