As datiLED states, soldering a wire to help pull out the driver is a good method. I've used it as well, successfully. Drivers are typically press-fit into the pill. The fit can be tight enough to make prying the driver's edge a hard task. Instead of soldering to the ground ring, however, I've made a wire loop by soldering to the center Batt+ pad. But it's all the same concept.
However, it's not clear to me how much help a driver swap will be if you simply go for a different 7135 linear regulator driver. The problem is that your battery source (I assume 1x18650 or similar) just can't deliver enough voltage to stay in full regulation when running in High. At 2.8A forward current, an SST-50 has a forward voltage of about 3.5V. IMO, that's way too high and it's no wonder your light fell out of High mode very quickly. At a minimum, your 1x18650 has to provide about 3.62V to reach and stay in full regulation. But that ignores additional voltage needed to overcome parasitic resistance from spring and battery contacts, switch contacts, etc. Even if the total parasitic resistance is say 0.1 ohms, that means another 0.28V, for a total of about 3.9V.
If you search CPF for HKJ's 18650 discharge graphs, you'll see that 1x18650 at 2A draw (not even 2.8A!) falls below 3.9V basically at the start, which is consistent with your observation that your light falls out of High in ~5 min.
Of course, the 3.9V estimate is just a guess, but the point is that it shows you what the challenge is -- a high voltage to stay in regulation due to a high LED Vf, an additional 0.12V overhead from the 7135, and parasitic resistance from the various contacts.
Switching to an XM-L will definitely help. An XM-L has a forward voltage of about 3.2V according to the Cree datasheet, so you save about 0.3V, which is a lot. Now maybe you need about 3.6V to reach and stay in regulation. I would guess that you might get 20 min or so of full power run time.
However, as I stated above, switching to another 7135 driver isn't going to help, since you haven't changed anything relative to the required input voltage (unless the 7135 driver is a lower-powered driver, such as one with 2.1A or even less of drive current for High mode).