I got one of these to try to remove slop from a hitch mount rack I use for a mini bike...I haven't had a chance to install it but seems like it would similarly solve your issue without drilling or having to weld something up...

MaxxHaul 50023 Hitch Tightener Anti Rattle Stabilizer for 1.25" to 2" Hitches Reduce Movement On Hitch Mount Cargo Carrier Bike Rack https://www.amazon.com/dp/B07JDD8L3H?ref_=cm_sw_r_apann_dp_FF44P9RA0C2ZD9SAXBQ6

Amazon. The specific one I got is no longer available and it was $35, a total steal.

But you can easily find sub $100 ones that are essentially the same design and specs: Hitch Carrier Kit

Having a good hitch tightener is a must to avoid road noise and wobble: Hitch Tightener