Have you checked the value of the cathode resistor on the 6CD6 yet, or read for cathode current? All this talk about serious mods so soon, and we're not even certain the flyback is the problem yet! Slow down, take a deep breath, do some trouble shooting. In the end if it ends up being the flyback, at least you did your homework first. All I'm saying is I definitely rule out every other possibility before jumping straight to unobtanium parts like flybacks, especially in a chassis like the CTC-2 where they rarely fail.
There are several controls on the chassis which impact HV production:
Drive varies the waveform going into the grid of the output tube, if it's low in amplitude the negative bias of the tube will not be right and tend to cause more current to be drawn heating things up. R175 (Sams) on my set had drifted causing incorrect operation of the drive control, I couldn't get rid of the drive line till I got the correct resistor in there.
Horizontal may be off frequency (wave coil), which will cause bad sync and low HV production- 2 of your original symptoms. The waveform in the oscillator itself may not be the right shape (equal peaks), the cure for both of these is to do the horizontal setup procedure in Sams.
HV pot may have drifted, if so the shunt may be dragging down HV even if everything else is working correctly.
You have a known bad convergence transformer- stop thinking about the fly till you cure that problem.
Is the CRT boot clean? If not, it will make a sizzling sound and even if it's not it could still have a path to ground somewhere that's draining the high potential away from the CRT. I went through 3 different ones before I found one my set 'liked'.
Or any one of a million other things on a very long list, but the point is you have to start somewhere before assuming the worst. My starting point is always to make sure the horizontal oscillator is in perfect health and on the right frequency FIRST. All other circuits in the chassis depend on horizontal to tell them when to do their thing: burst keying, color killer, convergence, ect. All other circuits are secondary to horizontal being correct.
My AGC pot was also bad, leading to a lengthy and unnecessary foray into CTC-2 IF alignment. Much was learned, including the important lesson of LOOKING FOR SIMPLE FIXES FIRST!
Slow down, maybe even walk away for a few days and think about it. Put John's convergence transformer in it, then retest. If it's still jacked up, go through the horizontal circuit with a fine toothed comb. Report to us with your results, the last thing we want is for your fly to be bad.