I agree that the days of selenium rectifiers are over, they were better than the copper oxide but that doesn't say much. Replace with silicon as a first step but even then small engine driven alternators perform very badly on single phase battery chargers with no inductance on the dc side. The conduction peak is only a short interval near the peak of the sine wave, by the time you include selenium rectifier loss, transformer impedance and the alternator reactance you are too far off the peak of the sine wave to get a decent mean current.
First step change the rectifier and I am sure there will be a fair improvement. If you have a variac and you have the potential to get 30% extra voltage from the charger you can try a low resistance inductor in the dc line between the rectifier and battery. This will spread the conduction angle over much more of the sine wave and the alternator reactance will have far less effect. This trick will make things worse unless you can get the extra voltage to change prom peak conduction to mean and you will be looking at something like a 24v output on the ac side of your transformer.
The big virtue of the selenium rectifier as pointed out was that the only test kit needed was your nose. I went to mend lots of TVs as a kid and when I met the small picture and the smell of the rectifier ( worse than cabbage in my opinion) I just went home for a new rectifier.
Flux