Convert Keras model to C++ using protobuf file and Tensorflow C API
My thoughts are to save Keras model to .pb(protobuf) file and then
load this model with Tensorflow C API.
Do you think this will be possible?

the answer is yes, absolutely. there are many ways to do this and i'll list some here so that you can pick up what suites your needs.

you can use cppflow, which is a c++ wrapper for the tensorflow C API, and actually makes it way easier to do the job than using the C API directly. they also have some examples on how to use it and in fact i think that this is the best solution out there.
you can see this wonderful repo hello_tf_c_api, especially this example.

or you can follow these articles on medium if you wish to use the c API directly:

undocumented tensorflow c api
deploying tensorflow 2.1 as c/c++ executable

there's also these two wonderful github gists that would walk you through how to do this exactly (although i recommend the c approach because it's more straightforward):

Training TensorFlow models in C
Training TensorFlow models in C++