Trev, I'd guess the gun is based on the Webley & Scott "Proprietary." A photo of the top of the gun would help clear that question up. The loop number looks like a W&S number from about that time frame. You can easily clear up these questions by buying the original sales records from Glasgow University Custodian of Archives.
As you are a reloader, you can easily make low pressure loads in 2 3/4" cases. Dig out a loading manual and find 7/8 oz to 1 oz loads that go about 1100 feet per second and have a chamber pressure of 5500 - 7500 psi. The low pressure will be kind to the metal parts and the low recoil will be kind the the 90 year old wood and to you. It is quite well established that low pressure 2 3/4" reloads don't have much pressure increase in 2 1/2" chambers. Many of us who shoot such guns regularly use this approach. Even if your gun had been rechmbered 2 3/4", it would not be suitable for USA - SAAMI standard loads as they are too high pressure. There are also plenty of factory specialty loads available via e-commerce and shipped to your door. Do not shoot loads of unknown pressure!!
The easy way to measure chamber length is to hold the barrels up towards a light source. you should see a very distinct shadow where the chamber meets the forcing cone. Slide a steel rule into the chamber until you see it touch the shadow. Read depth off rule. The depth measuring rod of a sliding caliper works well used in the same fashion as the rule above.
I doubt that the chamber have been lengthened
