Are you really sure you need your big numbers to be useable by old functions written with normal numbers in mind (that use the traditional operators)? If you only need the overloading for your own convenience on functions you cave control over you might be able to get by by using a different, custom operator for bignums.
For example, you could write a compiler to safely convert
var d = a <+> b <*> c;
into
var d = (a).add((b).multiply(c));
or perhaps, if you want automatic conversions...
var d = toBignum(a).add(toBignum(b).multiply(toBignum(c)));
I don't really see you being able to force overloading on an existing implementation without a big hassle. While you could teoretically replace all occurences of + with <+> and so on, current Javascript implementations are not optimized for this and I don't even want to start thinking what would happen if you tried to pass a bignum to one of the native C++ functions that are under the hood.
edit: Not needing to override the base integer class is the important point here (note how in the link you gave, overriding is the first thing the guy wants...). I don't think you will be able to find some kind "magic" optimization though as you have no control over which of the many JS implementations is being used by the client.
If you really don't like a custom operator like <+> the only way to distinguish a normal + operator (you shoudln't meddle with) from a fancy + operator (you want to do bignum stuff with) would be forcing some kind of ad-hoc typing system on top of Javascript, perhaps via comments, custom syntax or (as mentioned in a comment) hungarian notation. Just settling for a custom operator name you hate less would be less hackish, IMO.