I really think you need to reinforce the frame. Sub frames are not all that expensive and better safe that sorry. However, I have run into guys that have been hauling a sidecar for years without one and have had no problems.  I would not be without one. My sidecar car weighs about the same as a Velorex...165 pounds. Try
www.dmcsidecars.com or
www.perrysmotorcycles.com both have kits you can add. Or you can make your own, would be glad to send pictures of mine......which was done by Perry’s.
Another thing to consider is modifying  the steering geometry with a leading link or modified triple tree. This is where it can get expensive. I would first try the sidecar rig with standard steering. Then if it proves ........to be hard..... modify the steering. Adding a leading link or special triple tree is like adding power steering.....it does make a difference. Mine has a modified triple tree. But again I would not be without one or the other.
I have had my sidecar about 4 years now..10,000 miles or so. I have not had any problems with the cycle. I just went over it the other day making sure bolts were tight.....looking for stress cracks..........no problems at all with the cycle. I can’t find where carrying the sidecar has had any kind of a adverse effect on the bike.... other than slowing it down. It will slow you down.
The sidecar is another story. I have a Steib tub riding in a shop made cradle. I got stress cracks in the sidecar on every bolt hole. So, welded  3/16 steel plate strips, about 6 inches wide, where things were bolted down. Damn if I didn’t start getting stress cracks just beyond the steel plates.
My best guess is the cradle needed to be made from stronger.....thicker steel. I think the frame flexes and the sheet metal of the sidecar is way to thin to handle any flexing.
So, I am getting ready to do some major reinforcing to both cradle and bottom of the sidecar. Whatever sidecar you decided to go with.....use their frame and tub....don’t mix. If you make you own.........is better to err with stronger...... rather than lighter. I love my sidecar rig....it is so much fun, versatile. I haul dogs, grand kids and they all love it. It is my errand runner. You will love it I think.