The Mordern Bajaj Chetak Rides One
The Bajaj (and all Vespas) should be transported on their wheels, not on the center or side stand, so that the scooter's shocks and springs absorb any jolts instead of bending the stand or the metal body. Get some flat strap webbing about 3 foot or so longer than the handle bars and tie small bowline loops in each end. Run the webbing strap around the hand grips in a manner that the
loops point to the front and the strap goes direct from hand grip to hand grip.

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If you don't like this homemade web strap, you can buy one as shown in the picture called a Canyon Dancer. 

Put a used scooter tire or car tire at the front of the trailer and use tie downs to pull the front wheel into this spare tire, compressing the front shock/link slightly. The front wheel is going nowhere now.

Use tie downs on each side of the rear (the Bajaj rear seat handgrips are a good location), again compressing the shocks slightly. Now the rear wheel can't go anywhere. Basically your 4 tiedowns should now make a "v" going out from each end of the scooter.

It is ok to leave oil in the scoot as long as it is secure and can not fall onto on the battery side. Before tying the scoot down, all gas should be drained from the gas tank and the bike should then be run to drain all gas out of the carb as well.

You should have no problems if you do the above. I have trailered scoots using this method thousands of miles.

The Modern Bajaj Chetak Rides On The Modern Bajaj Chetak Rides On