Yes, it is possible to do a bike trip to Bhutan. You need atleast 10 days in hand to completely explore Bhutan. A possible itinerary would be - Day 1 - Arrival in Bagdogra, Drive from Bagdogra to Siliguri, Ride from Siliguri to Phuentsholling Day 2 - Riding from Phuentsholling to Thimphu (Via Chunka) Day 3 - Thimphu City Tour Day 4 - Riding from Thimphu - Punakha/Wangdue Phodrong (Via Do Chula Pass) Day 5 - Ride from Punakha/Wangdue Phodrong to Trongsa Day 6 - Ride from Trongsa back to Punakha/Wangdue Phodrong Day 7 - Punakha/Wangdue Phodrong - Paro Day 8 - Paro City Tours Day 9 - Paro – Phuentsholling Day 10 - Phuentsholling – Siliguri – Bagdogra.
Yes it is possible. Just keep your documents handy. Go to Phuensoling from Jaigaon. Get your visa and permit. 5-6 days including Thimphu, Punakha and Paro should suffice.