I've had the same problem but knew it was as a consequence of damage, just struggled to find it. My altered got out of shape in the traps, I wrongly stomped on the brakes and ended up bouncing around and landed from a considerable height. I hit the trans pan off the floor which wiped out the belly pan, broke motor mounts, and flattened chassis rails.
I fixed all the damage and re-installed the front brakes which is what caused the issue. I then had the same problem as you, vibration and breaking traction at 1000ft. It turned out bouncing around had buckled a wheel and bent a 35 spline half shaft. When chucked up, the shaft had about 2mm eccentricity. I replaced the shaft and wheels and it ran properly again and I had confidence in the car once more.
If the pan in your car is so low it's damaged the trailer floor, you will almost certainly be hitting the ground in the shutdown. Repeated hammering of the mounts will cause breakages like that.
With regards to tires. Don't go adding huge amounts of weight to balance them. Bubble balance is fine but balance the rim first and mark the heavy spot then fit the tire. Rotate the tire on the rim to offset any disparity between wheel and tire balance. Then add the minimum amount of weight to get them balanced. I don't use tape, I prefer to silicone the weights on, tape always lifts for me.
One more thing on the new car. If I were you, I'd go over the entire car and check out every weld, nut and bolt. Check the steering geometry, play in rod ends / thread engagement, tightness of jam nuts, play in steering / steering box etc. Put right anything you don't like
