NOTE: I completely missed the fact that you are working on a Bristol. I'm totally wrong about suggesting the Korky replacement, because it's not for a one-piece. I will leave the comments below in case somebody with a two-piece has a similar issue.
You shouldn't need more water to accomplish the old flush. Jim is right that you need to make sure you're starting with the right amount of water in the bowl. Refill hose is a likely source of the problem.
If you DO have everything set up correctly and you feel like you need more water in the flush, go back to the old flush valve, either with a Toto, unnecessarily-expensive, or this Korky:
https://www.korky.com/products/flus...ts/large-3-4-flush-valve-tank-bowl-gasket-kit
Available in a lot of places. Korky makes flappers and fill valves for Toto, here in America, in Wisconsin. The flapper on that valve is adjustable, so if you need to you can adjust it to stay open longer.
Korky sent me a sample of this kit a few years ago, and it works as advertised.