Here is what Rule 10 for the tournament says:
10. EQUIPMENT
Bowling Bags
No “homemade†or “modified†bowling ball carrying equipment is allowed to be used.
The tournament Manager on Duty will handle any questionable circumstances.
Since te handtruck is not manufactured as bowling equipment, I suspect that it may be forbidden. I really don't see the need for it anyway. In the past,I have taken 8 balls a 4-ball roller, and two 2-ball totes. I put one tote on top of the roller and carry the other one, so I really don't see the need for any additional carts.