I suppose there are multiple cases here:
If an ordinary statement ends with such a place name, should it be followed by a full stop/period? "He went to Westward Ho!" or "He went to Westward Ho!." The latter, though ugly, seems strictly correct to me and at least makes it clear that the sentence is not an exclamation, even if the reader has never heard of the place name.
In an exclamation, should an extra exclamation mark be added? e.g. "He went to Westward Ho!!" Is this, in fact, a restricted situation where multiple exclamation marks could legitimately occur together?
In a question, I do not see any alternative to simply following the original exclamation mark with a question mark, although it looks like an interrobang: "Did he go to Westward Ho!?"
Finally, just for completeness and to consider the most unpleasant case, suppose one wanted to express incredulity using an interrobang. Would one then write "He went to Westward Ho!!?"
(Naturally, I suspect that in some of these cases, unless one were quoting, the best advice would be "Recast the sentence".)