This is not a complete answer as I do not have SageMaker setup (And I do not know MXNet) and so I can not practically test this approach (yes, as already mentioned, I do not want to call this a complete answer rather a probable pointer/approach to solve this issue).
The Assumption -
You mentioned a that your model is very similar to the notebook link you provided. If you read the text in the notebook carefully, you will see at some point there is something like this -
"In this demo, we are using Caltech-256 dataset, which contains 30608 images of 256 objects. For the training and validation data, we follow the splitting scheme in this MXNet example."
See the mention of MXNet there? Let us assume that you did not change a lot and hence your model is built using MXNet as well.
The Approach -
Assuming what I just mentioned, if you go and search in the documentation of AWS SageMaker Python SDK you will see a section about serialization of the modules. Which again, by itself, starts with another assumption -
"If you train function returns a Module object, it will be serialized by the default Module serialization system, unless you've specified a custom save function."
Assuming that this is True for your case, further reading in the same document tells us that "model-shapes.json" is a JSON serialised representation of your models, "model-symbol.json" is the serialization of the module symbols created by calling the 'save' function on the 'symbol' property of module, and finally "module.params" is the serialized (I am not sure if it is text or binary format) form of the module parameters.
Equipped with this knowledge we go and look into the documentation of MXNet. And Voila! We see here how we can save and load models with MXNet. So as you already have those saved files, you just need to load them in a local installation of MXNet and then run them to predict the unknown.
I hope this will help you to find a direction to solve your problem.
Bonus -
I am not sure if this also can do the same job, (it is also mentioned by @Seth Rothschild in the comments) but it should, you can see that AWS SageMaker Python SDK has a way to load models from saved ones as well.