Check out algorithmia - if that works for you.

Instead of looking for ML specific services, look at docker or container options. Its better if they have instance warm ups that way you can launch the vm when necessary.

Also see if you can split your solution into 2 parts. One that gathers intelligence in an offline batch mode. This data should help the second part to respond faster. This would work with some Algos only. Not so much for Deep learning

Explore deep learning compression techniques to reduce size of the model.

There ain't many services for one off models. It doesn't make economical sense for cloud providers.