Couple quick items for you.
Yes, it is possible for the weakened frame section to be affecting the ride, but if in fact those bolted on sections are structural and not cosmetic, then you might have something else to look at.
You CAN go to a Weld shop and have them weld just about anything as long as there is some good metal for them to weld to, however, if you're talking the cross member you might be better off finding one in a boneyard. If however you are talking the front to rear members, then the only way is to have the weld shop weld new structural metal into it.
I would suggest you check a couple other things out first, (although don't ignore that rust, get it fixed asap).
First off, are you sure the tires are balanced properly? That is, with the SPIN balancer and not the backyard mechanics bubble level. Also are they balanced to center axis? Up here in the Northwest, there is an outfit called Les Schwab, they do an awesome job of balancing not only for radial balance, but also for Axial balance, (i.e. inside edge versus outside edge). A poorly balanced tire ("that's close enough") may in fact exhibit the rattling shaking you are experiencing above 40mph. My wife's car was balanced to "almost perfect" but at 65 to 75 it would shake the steering wheel, not enough to loose control, but enough to make you slow down to avoid it. Yet there were times when you wouldn't notice it.
The other item is BELT SEPARATION. I had a set of Michelin tires on a 76 Mercury Capri II some years ago, that had a broken belt in each of the front tires. At 30 mph the whole car would shake and bounce, at 40 it would level off and be ok, and then it would pick up again at 55 or so, and drop off about 75. When I finally took it in to have the car serviced, the first thing they pointed out as needing to be done, was NEW tires. Mind you, there were times when you wouldn't notice it.
Lastly, check to make sure your front end is aligned, and also check to make sure that the front of the car has been aligned to the rear of the car. This is called dog-tracking. A car that's been hit, or had the rear-end completely worked on, or the front end, needs to make sure the front tires are in line and parallel to the rear tires. If they're not, you'll dog-track and in severe cases you'll have some of the shaking you wrote about.