A clean sheet engine in the mid/late 30s would be better than any major effort. The Perseus was that attempt except it didn't work out as planned.
Anything can be improved if you spend enough time and money on it. The Question is should you or would the time and money be better spent elsewhere.
The Pegasus had some problems that were due to it's age.
However a lot of engine design is a compromise.
A lot of engine designs were the way they were do to the materials and manufacturing ablities of the times.
WW I engines mostly had narrow bores and long strokes. It was more than just fashion. There were limits on how big you could make the piston and still have the middle stay cool (not melt). With plain steel valves there limits on how big you could make the valves and not have them overheat or warp and not seal which lead to failure very quickly. Valve springs often broke even at 2000rpm or less. And a bunch of other problems.
Money was often scarce in the 1920s and at least the first 1/2 of the 30s. Aircraft engine companies very often tried to save money buy using existing tooling or parts on several different engines. This sort of worked in the 20s with low rpm engines (around 2000rpm, give or take). For Bristol using 3 Mercury size cylinders (or short Jupiter) for a 120-140hp engine looked attractive to the accountants. Lots of common parts and a way to get into the small aircraft market. But a 3 cylinder 8 liter engine was both not comfortable to fly behind, they tended to vibrate things loose in the aircraft structure. What was saved in engine maintenance was canceled out by airframe maintenance. There were 5 & 7 cylinder engines using the same cylinders although this saw a little more success French or Italian license production?
Bit of sidetrack there but many companies held on to existing tooling and designs too long or tried to make new designs fit old tooling which limited the new designs. Wright had merged with Curtiss and at the time the corporation had their own airline/s. Which meant they had their own market for their own engines. But even Curtiss-Wright didn't have enough money to do everything, in the early 30s they had at least 6 different engines in two different factories and finally decided to simplify things after the merger. Curtiss had been a leader in liquid cooled engine design in the 20s but they were limited in sales by the surplus Liberty engines in several ways. Why buy a new 400-440hp Curtiss D-12 when you can buy an in crate, brand new 400hp surplus Liberty for a fraction of the price? Turns out many of of Liberties (and early radiators) leaked water so bad than many US military aviators, once they had moved up in rank, swore never to buy liquid cooled engines again
With the Market disappearing for the D-12 and the 25.7 liter Conqueror C&W shifted resources into the R-1820E (enlarged and improved R-1750) and just trickled improvements to the Whirlwind 5,7,9 cylinder engines. This market shift plus the loss of their chief designer stopped development and sales of Packard liquid cooled engines for nearly 10 years which meant that Wright only had to compete with P&W for the US market and many foreign markets.
Again a side track but it explains why C&W had some of the money or were willing to invest in rapid engine development in the 30s.
The 1920s had a lot of different ideas floating around. Wright had the experimental P-1 floating around in 1923/23 with 6in X 6.5in cylinders 1654cu in (27liters) but the cylinder head, while using very sloped valves uses a rather strange pushrod/crank arrangement that was soon changed. In was in the mid 20s that Radials began to use "mixing fans", very low boost superchargers that mainly assured a somewhat equal amount of fuel/air mix went to all cylinders. Long skinny cylinders and more cooling surface for the volume and short distance from center of piston to the cylinder walls. But skinny cylinders restrict valve area. Then we hit the problems of flame travel in the cylinder and scavenging. Theory says that the fuel burn should be most completed when the piston is about 20 degree past TDC for efficiency. Expanding burnt gases push the piston the rest of the way down. Everybody was closer to 5 to 1 compression ratio than they were to 6 to 1 so this was important. I haven't done the math to see if long skinny cylinders had less scrubbed surface (area the piston/rings rub over each stroke, roughly 80% of the internal friction in an engine) than short fat ones, may depend on exact measurements. But the long stroke introduces the increase acceleration loads at higher rpm( pistons have to start/stop and each end of the stroke and the accelerate to high speed and slow down again). Forces are usually considered to go up with the square of the rpm, a real reason that C&W kept changing the crankcases and shafts every time they increased the RPM.
The 4 valve head Bristol used was a good idea at the time, 4 small valves gave more area than 2 if they were all vertical or nearly so in the cylinder. 4 small valves were easier to keep cool and didn't warp as much as two larger ones. But this was with the early 20s valve materials and valve seat materials and even with the head materials and casting/forging techniques. A lot of the new materials were introduced over 3-5 years so there was not a single "AH-HA" moment. It even took well into the 30s to get to really good valve seat materials. Wright and P&W had bet in the mid/late 20s on shorter stroke (still had longer stroke than bore, just not as skinny as the British) and two valve heads with a lot of angle between the valves. Might have been failures if the Salt and Sodium filled exhaust valves had not worked out.
Now I have not addressed (because I don't know) any problems the Bristol engines may have had with oil or bearing failures. We do know that some engines were not supposed to be flown at certain rpm bands due to vibration but some Wright and P&W engines had similar restrictions. Sometimes later versions fixed that (new vibration dampers?)
By 1943/44 the Wright R-1820 was running at 2800rpm and had a slightly lower piston speed than the Pegasus did running at 2600rpm. It is not quite as bad as that because the fatter/heavier piston of the R-1820 create more load but you get the idea. I will also repeat that Wright shortened the stroke (slightly) of their 14 & 18 cylinder engines to reduce the piston speed at the higher rpms. Post war Wright just used 7 cylinders from an R-2600 to make a R-1300 to fill in that part of their product line. Most model changes had to do with vibration dampeners.
A lot of companies liked to stick with old dimensions not just because of existing tooling but in the days before computers figuring out vibration problems took a lot of work, both doing calculations and running experiments to see they were right. Vibration problems go up exponentially with the rpm. Engine might be fine 2400rpm, going to 2600rpm introduces a harmonic that breaks the crankshaft in short order.
Story about the First Griffon engines. They rotate backwards from Merlins. They broke the first two crankshafts in short order. Then somebody noticed (or said out loud) that instead of making the crankshafts the same as a Merlin and trying to run them backward, they make the crankshafts like a mirror image of the Merlin crankshaft with the crank throws doing the mirror image. They tried it and the new crank ran for hundreds of hours. They had only been studying vibration problems for around 20 years at the start of WW II. In WW I they were just worried about getting the engine to run at all for 20-30 hours.
I don't have a real good idea if Bristol could have improved the Pegasus substantially by throwing enough money at it. There are certainly theoretical problems with it's basic design that needed changing to make it easier to get improvements. But a lot of those improvements also require substantial sums of money, like new, shorter, fatter cylinders with much improved fins. New cylinder heads with more fins (new valve gear?) improved bottom end, and so on. May keep the Pegasus name but new engine.
