The distinction between "Transition" and "Post Transition" is so nebulous that I've dropped it. It's more accurate to use Family Era and Corporate Era. Here's why:
Finlay bought the family business in October of 1960, but retained Montague Barling and Williamson-Barling to run it. The family continued to run the business after it was sold for about 20+ months. The markings on the pipes did not change until mid 1962. So a lot of "Pre-Transition" pipes are technically "Transition" pipes because they were made after the sale of the company to Finlay, the so called "Transition" company. People think that they're Pre_trans" pipe because the markings are the same. You have 20+ months of porduction of "Pre-Transition" pipes that are technically bogus if one uses the current "Pre-Trans", "Trans", and "Post-Trans" definition.
In mid 1962 Barling releases it's 150th Anniversary Catalog which shows that the numbering system has changed. We now have a 4 digit numbering system wherein the first digit ranges from 2-6. There are no 4 digit model numbers beginning with a 1. The old "Barling's Make" logo is retained. The family is still running the Business for the new owners at this point, but not for much longer. In June 1962, Williamson-Barling gets the sack. He will later run Charatan. Montague steps down as President shortly thereafter, and Finlay brings in their own management.
Finlay had been partially owned by the Imperial Tobacco Group long before it bought Barling. Imperial had a 49% stake in Finlay with an option to buy the rest of the company at their discretion. This they did in February of 1963. Imperial owned Barling starting in early 1963.
To the extent that there is a "Transition" period, one that exists after the family is gone, it lasts for about 7-8 months. Trying to tag a Barling as "Transition" VS "Post-Transition" is not a 100% possible, any more than trying to Tag a Barling as "Pre-Transition" vs "Transition" is 100% possible. The old definitions were based on incorrect assumptions and are just not an accurate depiction of the actual history. More accurate to say "Family Era" vs "Corporate Era".
All that said, Here's what I can say about your pipe. It was made between late 1962 and 1970 as it was certainly made in one of the two Barling factories. The factories where shut down around 1970 and the work was then outsourced to several other pipemakers, such as Charatan. Judging from the shaping of both the bowl and the stem I'd put this at the earlier end of the range. I have several Barlings from this period and they are uniformly excellent performers. It should smoke like a champ.