It's exciting to hear your students are asking such great questions. A short answer to your question is "yes" if someone naively approached your graph they would need to try all possible paths while you would immediately know the correct answer. Unfortunately this doesn't make for a very secure protocol since as students learned from the lesson on the Vigenere Cipher, typically the methods (though not the specific keys) used for encryption are known publicly. In the case you're describing someone cracking your encryption would know to just sort the edge weights from smallest to largest and then pick the smallest edges in order to form the shortest cycle. In fact you would know with certainty this is the shortest cycle since there would be no shorter set of edges you could have used.
Your next instinct might be to slightly obscure the shortest path by adding a few edges shorter than the ones used in your cycle. Unfortunately now you're probably not sure that you have the shortest cycle anymore and would need to check with, you guessed it, a brute force search.
Let me know if you want to talk more, thanks for sharing this great question, and I hope things are going well.