I don't know that this will answer any questions but perhaps it will raise a few about the W-R 'T' prefixed guns.
This is a partial photograph of a page of Westley Richards shotguns in inventory at Abercrombie & Fitch as of February 1, 1929. The page was created at the time when A&F had purchased Von Lengerke & Detmold, its Manhattan competitor, and the inventories of the two stores were combined in one ledger. There are 16 shotguns in the abbreviated list - 12 'T' prefixed guns, 2 letter 'O' prefixed guns, and two guns with no letter prefix. Most of the 'T' prefixed guns have serial numbers that start with T96xx. (eight to be exact). These numbers generally correspond to the serial number date range for the 'T' prefixed guns that is to be found in Nigel Brown's "British Gunmakers - Vol. 2". What is interesting is that the model name given to the 'T' guns is "Federal Grade". I have searched the VL&D, VL&A, and A&F catalogs between 1926 and 1933 and I do not even find the Westley Richards shotguns listed even though many other British makes are described. They seem to have had enough on hand to have put them in the catalog.
Of further interest, of the 12 T guns, three are a 16 ga, one is a 20 ga, and the rest are 12's. About half are full pistol grip (F), a couple are half pistol grip (1/2), and the rest are straight grips (STR). One of the T guns is marked as having a detachable lock (DET LOCK) and several as having single triggers (S.T.). This seems an anomaly if the guns are of a utility grade. The price of the DT guns was $300 and the ST guns were $390. The detachable lock T gun was $475. In 1929, a CHE Parker was only a bit over $200 so these were not inexpensive guns.
I also checked the 1907 to 1909 VL&D book and found several T guns there as well, including several single barrel guns. Most had serial numbers that began T28xx which also agrees with the Nigel Brown list. (Interestingly, one Westley Richards that did not have a T prefixed number was sold to John Philip Sousa.) There are four different models of W-R shotguns listed in their 1908 catalog but there is no indication which of the letter prefixed guns each may be, if any.
Sorry if this rambled a bit but perhaps it will help in some way.