Luke:
You will get a lot of opinions so here is mine based on my experience as a heat transfer engineer.
The fluctuations and problems (except the high CHT on one cylinder in climb) come from fuel boiling- in the lines feeding the fuel pump, in the fuel pump, between fuel pump and "spider," and in the "spider" assembly and the little lines where the fuel spreads out to the cylinders. The boiling leads to intermittent flow, bubbles, more flow, and its occurrence depends primarily on temperature and pressure, both of which change with time and RPM after start. And the rate of cooling of each of these areas is different once the engine starts, so you can have boiling occuring I several places at once which greatly confuses the situation and make engine operation even more erratic after a hot start.
Depending on the distance between tank fuel elevation (empty or full?) and fuel pump elevation and length of line under the cowl leading to the fuel pump, lowered pressure caused by fuel pump suction will cause bubbling (boiling) of fuel in the inlet line with consequent erratic flow at the fuel pump outlet as bubbles are swallowed, clear, and more are swallowed. This will also happen in the gascolator if it is mounted in front of the firewall. Same thing happens downstream of the pump where fuel boils in the lines due to elevated temperatures after shut down.
And consider this scenario. You start the engine. RPM rises, fuel demand increases, the fuel pump sucks harder, fuel boils at the inlet, pump sucks gas vapor, fuel pressure falls, then fuel pumped above the engine for a moment starts to boil when it hits the hot lines and the fuel pressure falls when the fuel pump falters. Suction falls due to vapor in the pump. Liquid finally gets to the pump inlet, the pump raises its outlet pressure, puts more fuel above the engine, and the cycle repeats with boiling at the fuel pump inlet and the fuel above the engine starts to boil again when the pressure falls again. The cycle repeats. Surge. And the fuel mixture? All over the map. This can continue until things cool off which can take 1-2 minutes, a very long time when you are battling an engine that is running out of fuel (vapor only), then drowning in it, then starvation sets in again with a new batch of bubbles.... You get the idea.
At higher RPM, the fuel pressure out of the pump rises to meet increased flow rate demand for the engine, and this raises the pump discharge pressure which can suppress boiling downstream of the pump, but may worsen boiling upstream of the pump due to decreased pump inlet pressure 9greater suction). Reduce the RPM down before the lines cool, and the boiling starts again downstream of the pump and fuel nozzles get squirts of fuel, then vapor, then fuel and this continues until no more vapor is produced as temperature falls due to fuel cooling of the lines.
Wrapping lines helps only for a short period after shutdown. The insulation on the lines is insufficient to keep the heat out of the lines and out of the fuel within the lines for more than a feww minutes. Thermal insulators are actually lousy insulators. Think of them as semiconductors, not insulators and conductors in the electrical sense where insulators really insulate. Remember: there is no fuel flow to carry any heat away the heat seeping through the insulation unlike occurs when the engine is running.
Under stagnant conditions after shutdown, temperatures tend to equilibrate fairly quickly. Under the cowl after shutdown, temperatures will soar to over 150F+ even on a cool day before they start down due to natural cooling which can be very slow with a tightly cowled engine and no wind.
Facing the airplane into the wind helps to bring air into the inlets and push heat out the outlets as occurs during flight. But without wind present, the natural convection after shutdown will be in the bottom, rising up through the cylinders, and out the inlets behind the prop. This will be a slow process because there is not much elevation difference between cylinders and "exit" and thus no "chimney" effect.
So the root cause is that ALL the fuel lines (and pump, and gascolator) under the cowl heat soak and then problems occur when the engine starts and needs a steady diet of liquid fuel at the injectors, and gets spurts of liquid, then vapor, then liquid, and it continues for several minutes.
The solutions offered to accelerate fuel line and fuel pump (and gascolator) cooling are beneficial: circulate cool fuel back to the tank to cool as much as the plumbing as possible before start (takes at least a minute, and only cools a portion of the lines), and opening an oil door other other means to let hot air escape out the top of the cowl to cool everything underneath. Otherwise, you have to start, and if you can keep it running a bit, increase RPM, perhaps more than a comfortable amount (like 1700, a lot at a tie down) to get steady operation (high pressure downstream of the pump), and flow lots of fuel under pressure until things cool down. That can take a couple of minutes or more.
Also, after starting, insulated fuel lines will cool less rapidly with air blast after starting than un-insulated fuel lines. I have in mind here the stainless steel small lines emanating from the spider to the fuel injectors. All others should be fire shielded.
On my Continental, both the gascolator and the fuel pump have shrouds and get blasts of cooling air via one inch SCAT lines fed from above the cowl. This eliminates heating in flight from the hot air downstream of the engine. A short amount of low boost prior to start gets fuel moving, and the engine usually starts promptly even with a very hot start, after which I shut off the low boost.
I measure air temperature at the face of the firewall, and temperatures of 150F are typical in cruise operating with CHT of 300F with mixture LOP and OAT of 60-70F in the summertime at 6000-8000. In other words, with a 300F CHT, I see temperature rise across the engine of about 130-140F at 65% power. Higher CHT and higher power settings will increase the temperature rise across the engine. This is the temperature at which everything is heat soaked downstream of the cylinders and on the firewall. Add more temperature rise if the component (fuel pump, fuel line, or gascolator) can "see" the thermal radiation from the exhaust pipes which are glowing red hot.
In short, there are lots of ways to heat soak fuel lines, pumps, gascolators, and such both in flight and after shutdown. All contribute to fuel boiling and erratic operation after a hot start.
The high CHT, being recent and apparently severe suggests to me a cracked spark plug causing pre-ignition which is extremely harmful and causes rapid excessive heating. Pull the plugs, have a close look, and if you can not find a crack, chip, or flaw even with a magnifying glass and a good light, I suggest two new plugs into that cylinder and a test flight based on general principles. Preignition and detonation are the quickest ways to convert cylinders and pistons to scrap - sometimes in mere tens of seconds. Seek the cause before next flight.
You may now insert two cents in that slot in the front of your computer.
Fred Moreno