This results in characters not showing up at all or totally different characters being shown instead.

The site is only displayed correct when you override the automatic selection and choose "Unicode (UTF-8)" manually.

Therefore I suggest you add a proper page header to all the pages that tells the web browser which encoding to use.

Here's an example:

<head>
<meta http-equiv="content-type" content="text/html; charset=ISO-8859-1">
</head>

See http://www.iana.org/assignments/character-setsfor valid charsets (utf-8 can be found there too).