Does the compilation process offer enough security such that a motivated thief wouldn't be able to identify the encoding/decoding techniques that are in use to protect the stored encrypted data?
No.
First off, you don't say what you really mean by "compilation process". There is no process with Javascript that is analogous to compiling C++ into binary assembly code. Javascript is an interpreted language, not a compiled language. The Javascript interpreter may do a compile step internal to the JS engine, but that's not something you can do yourself. The node.js Javascript engine requires plain text legal Javascript as an input.
So, any compilation process you would run on your Javascript code would just be compacting and perhaps obscuring it. It's still plain text Javascript code. It's still runnable by anyone. Anyone can still diagnose what it does or how it works. Obscuring it (like replacing descriptive variable names with short one or two letter names) makes it a little more work to analyze the code and understand it, but it is only a temporary obstacle that can still be overcome by any determined hacker.
If someone were ever able to steal that server they could get access
to the source code. They would have the keys and decoding algorithms
to our encoded data.
You will need to physically secure access to your server and the code on that server in order to protect it and you will need to rely on that protection.
If you are using a packaging tool that creates a runnable .exe file, then keep in mind that that is not really compiling your Javascript. It's just packing it into a shell of a .exe so that the native code .exe can run, can extract your Javascript from within the .exe, can put that in a temp file on the hard disk and can then execute node.js passing it that Javascript source file. This is a packaging step, not a binary compilation step. Now, the package in the EXE may be binary compressed, but when the code is passed to node.exe it's still going to be plain text Javascript which can still be seen by a determined hacker.