The issue is not one of showing a native name plus an English-language exonym. For the time being, the lake has two native, English-language names. Neither name is an exonym. Lake Ontario is not a Canadian lake being called by a different name by the USA.
Now suppose Tramp decreed that The Great Bear Lake in Canada should be called Star-Spangled Lake, effectively making that the USA's exonym for a place in Canada. And then suppose Google showed that name to visitors outside the USA, along side The Great Bear Lake. That would then be like (or even worse than) showing English-languge exonyms for German places to someone whose language is not indicated as English.
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The issue is not one of showing a native name plus an English-language exonym. For the time being, the lake has two native, English-language names. Neither name is an exonym. Lake Ontario is not a Canadian lake being called by a different name by the USA.
Now suppose Tramp decreed that The Great Bear Lake in Canada should be called Star-Spangled Lake, effectively making that the USA's exonym for a place in Canada. And then suppose Google showed that name to visitors outside the USA, along side The Great Bear Lake. That would then be like (or even worse than) showing English-languge exonyms for German places to someone whose language is not indicated as English.