Putting a resistor across it was what I was thinking to do. I wrote a long post about it, but I was thinking about it in the way that increasing the resistance would increase the output of the video amp tube, but then I realized that that's completely wrong - it's the opposite, decreasing the resistance increases the output. So I deleted my post quickly, hoping not to look like a total fool.
So adding the resistor across it would only keep the output of the vid amp tube, and hence the signal entering the 1st sync tube, from getting too low - it would lower the max resistance, when the control was all the way out it would be like 10k in parallel with whatever else I put in there.
But on the other end the min resistance would still be 0. It wouldn't solve my original problem where the p-p voltage of the signal entering the grid of the 1st sync tube was too high with the contrast control cranked up. But if keeping my extra capacitor in there, in addition to this resistor will let it sync no matter where I put the contrast control, I could just leave it in there. Or I suppose I could also put a resistor in series with the control as well and really try to limit it's range.
The AGC isn't adjustable.