The problem is your assumption that it will try a dynamic implicit convert on arguments of an dynamic invocation to make a method call work, this is not true.
When your arguments aren't statically typed, it will use the runtime type to find the best matching method (if the runtime type matches the static rules for implicit conversion to the argument type this will work too), since your your IntProperty,StringProperty seem to be returning a DynamicObject rather than an Int and a String or something that could statically be converter implicitly, this lookup will fail.
If SomeDynamicObjectImplementer could actually return an Int for IntProperty and a String for StringProperty your method call for without casting would actually work. It's also probably a better dynamic typing practice if you data type is based on the actually type of data rather than usage using try convert. You could add actually implicit convert methods for every possible type that you could return to that returned DynamicObject type, but that could cause strange resolution issues to depending on how much you are overloading.
However, another option to keep your dynamic implementation the same is to mix a little controlled static typing in, you can use ImpromputInterface (in nuget) to put an interface on top of a dynamic object, if you do that then the TryConvert method would be called on your returned DynamicObjects.
public interface ISomeStaticInterface{
int IntProperty {get;}
string StringProperty {get;}
}
...
var myVar = new SomeDynamicObjectImplementer().ActLike<ISomeStaticInterface>();
method(myVar.IntProperty, myVar.StringProperty);