For instance, I want to do something like the following:
char output = 'A' + 1;
getting output of 'B'
However, simply trying what I have above gives a casting error.

Yes, the binary + operator isn't defined on char, so you end up with implicit conversions to int. You can use a pre/post-increment operator:
char x = 'A';
x++;
... because the ++ operator is defined for char.
But otherwise, you need a cast:
char x = 'A';
x = (char) (x + 1);
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