UPDATED
There's no need to create a custom HttpMessageConverter since AbstractHttpMessageConverter has a method setSupportedMediaTypes which can be used to change supported media type:
MappingJackson2HttpMessageConverter converter = new MappingJackson2HttpMessageConverter();
converter.setSupportedMediaTypes(Arrays.asList(MediaType.TEXT_PLAIN, MediaType.APPLICATION_JSON));
restTemplate.getMessageConverters().add(0, converter);
I think it's possible via implementing own HttpMessageConverter<T>. 
RestTemplate uses it to convert a raw response to some representation (for instance, POJO). Since it has a list of converters, it finds specific converter for a particular response by its type (e.g application/json, etc).
So your implementation of HttpMessageConverter<T> should be something like default MappingJackson2HttpMessageConverter but with changed supported media type:
public class MappingJackson2HttpMessageConverter2 extends AbstractJackson2HttpMessageConverter {
    private String jsonPrefix;
    public MappingJackson2HttpMessageConverter2() {
        this(Jackson2ObjectMapperBuilder.json().build());
    }
    public MappingJackson2HttpMessageConverter2(ObjectMapper objectMapper) {
        // here changed media type
        super(objectMapper, MediaType.TEXT_PLAIN);
    }
    public void setJsonPrefix(String jsonPrefix) {
        this.jsonPrefix = jsonPrefix;
    }
    public void setPrefixJson(boolean prefixJson) {
        this.jsonPrefix = (prefixJson ? ")]}', " : null);
    }
    @Override
    protected void writePrefix(JsonGenerator generator, Object object) throws IOException {
        if (this.jsonPrefix != null) {
            generator.writeRaw(this.jsonPrefix);
        }
        String jsonpFunction =
                (object instanceof MappingJacksonValue ? ((MappingJacksonValue) object).getJsonpFunction() : null);
        if (jsonpFunction != null) {
            generator.writeRaw("/**/");
            generator.writeRaw(jsonpFunction + "(");
        }
    }
    @Override
    protected void writeSuffix(JsonGenerator generator, Object object) throws IOException {
        String jsonpFunction =
                (object instanceof MappingJacksonValue ? ((MappingJacksonValue) object).getJsonpFunction() : null);
        if (jsonpFunction != null) {
            generator.writeRaw(");");
        }
    }
}
Then you can add this to RestTemplate object:
restTemplate.getMessageConverters().add(0, new MappingJackson2HttpMessageConverter2());