Protecting diode bridge from higher current
If you are confident that the transformer output is appropriate for charging your battery, one solution would be to simply increase the current-carrying capacity of your bridge rectifier.  

You mention that you can't obtain higher current diode bridges. Can you obtain higher current discrete diodes?  If not, and your application can handle the cost of 2 diode bridges, you can make an effective 20A-rated full bridge rectifier out of two 10A full diode bridge modules.

If you look at the schematic for a full bridge rectifier, you'll notice that it is possible to connect the two AC input terminals together to obtain two diodes in parallel between the AC input and each of + and -.  Since the diodes in a bridge rectifier are usually on the same die, they are likely to be reasonably well matched and likely to current share reasonably well (though never perfectly, so some derating is warranted).   Simply use two full bridges connected in this way (one bridge for each AC input with the rectifier + and - terminals each connected in parallel)