I changed the circuit to this.
View attachment 146305
Hi,
Well to start, transformers used with rectifier circuits have to be derated. Using a 10 amp transformer with a 9 amp load is probably too close.
One way to look at this is because of the harmonics, but the simpler way is to realize that the current waveshape of a capacitor filtered full wave rectifier circuit is very different from a pure sine wave. The capacitor charges for a much shorter time than would be present if there was a sinusoidal current waveshape.
For example, if the capacitor voltage was 100v then the diodes would not conduct again until the input voltage got up to 100v. That results in a current waveshape that is very narrow with a high peak. You can easily see this if you do a simulation which I highly recommend, or someone else does it and shows the results here.
The transformer heating would be different for this kind of waveshape than for a sine wave, and transformers are usually rated in terms of waveshape.
This was known a very very long time ago, and that is why inductor input filters became in vogue. The inductor causes the diodes to conduct over a larger span of the input wave and so the waveshape of the transformer current is more sinusoidal again. One solution then is to add an inductor in series with the secondary. This is not that simple either though because the inductor might have to be rather large, but you can look into this.
You can instead add an inductor at the output of the rectifier, but then it has to be able to handle the full DC current.
Another solution may be to see if you can just reduce the capacitance. Of course that means less filtering, while the inductor addition actually means better filtering.
Another possibility is to add a high velocity fan to cool the variac. Maybe a regular computer fan would do, but you may need a better fan than that. Look at the power use of the fan it may have to be as high as 20 watts or so. These are more expensive and make more noise, but they blow a lot of air.