Answered on June 07,2019
As you are staying in Bangalore for shorter duration, you dont need to pay road tax in Bangalore. If you are staying in Bangalore for more than 11 months, then you need to pay road tax and get your vehicle re-registered.
As you keep on roaming, it is better to keep bills at toll plaza near the Karnataka border with you. This will help if cops catch you for anything related to road tax.
Answered on November 11,2019
You can check this video to calculate the road tax to be paid in Bangalore.