Hi Vishnu! I think train and bus are the only cheapest way to reach Bhutan. Its time consuming but yes, if you are planning a trip under budget this is how you can travel to Bhutan.
The only way which would be cheap & most convenient for you is to board a train from Coimbatore to New Jalpaiguri. And then get a taxi from there. Remember to pre book the package as you have to get a permit before getting into bhutan.
For more info please dm me. Happy Travelling!!