Apologies if I'm teaching you to suck eggs, but I'd start by making sure everything's set up correctly and not worn out.
Decoke the head and lap the valves in, then refit the head with a new gasket and check your compression - if it's below 120psi across the board or there's more than about 10psi difference between any cylinder it'll need a rebuild.
Assuming it's okay, fit a new cam chain and make absolutely sure your cam timing and valve clearances are perfect. Fit new or rebuilt carbs with heat shields (to prevent fuel evaporation) and a new or reconditioned distributor together with an electronic ignition module, new coil, leads and plugs. Set the ignition timing to 6 degrees btdc, fire it up and see how it feels.
Assuming everything is working as it should and you still want more power, I managed that with a Mk2 Vitesse cam and 1.75 Strombergs on adaptors. My motor had been going flat at 4500rpm and returning less than 20mpg, but those mods freed up the revs and gave around a 50% improvement in fuel economy to boot. It still wasn't a ball of fire but it was quicker and much more fun to drive.
For track use you really need to go large and/or modify. Phil J's green Spyder runs a 2.5 TR5 motor with a hot cam and six bike carbs. It pushes out close to 200bhp and has been clocked on the track at over 140mph (with 3.63 diff and overdrive). It also lapped faster than both a Porsche 911 and a Lotus 11, whose owner had laughed at Phil's car telling him the aerodynamics would make it too slow for him to compete with his Lotus.
I believe Phil posted on here that he could put another six carb setup together if anyone wanted one, but that was a while ago so you should pm him if you're interested.
Unless they're worn out, just changing the carbs for bigger ones or more of them won't make a dramatic difference without changing the cam and modifying the head. Witor would be a good starting point for parts and advice (other Triumph specialists are available) -
https://www.chriswitor.com/products.php?cat=254&pg=2
Good luck and keep on enjoying your car.