I don't know what the approach was to upgrade this car, but perhaps by comparing against steps taken in another approach that yields good results, it may be possible to gain some insight and go from there.
A stock '84 Spider has a soft suspension (soft springs, SPICA shocks), sway bars in the front and rear, 14"x6" light alloy wheels (12 lbs. each), and touring type "H rated" tires. It yields a "soft" ride, and the nose dives when one breaks hard, but the handling in a straight line or through a corner, in dry conditions or in heavy rain, is "balanced".
Typically, the first handling upgrade is to replace the wheels and tires. (This yields the biggest "bang for the buck".) Going to a 16" wheel, with modern tires should yield a noticeable improvement. It's really hard to imagine that anyone can go wrong with this upgrade, unless different size tires where used. For example, if the rear tires were wider than the front tires, this yield an under steering car. This would be very noticeable to a driver and can be easily corrected before going to the next upgrade.
The next upgrade would be to stiffen the suspension. There are many proven sets sold by suppliers, and there are also combinations that racers use. One can't go wrong with the sets, as these are the result of several evolutions and the R&D to achieve the required balance has been tested with real cars. (The work done by W&D to develop their street and race springs is a good example.) One can also buy springs independently and fit them to the car, as some of the racers do. However, they typically go through several combinations before achieving the balance they are looking for, unless they are following someone else's proven combination for the specific car. (Lets' remember that there are not so insignificant weight differences between Spiders of different generations.) Bottom line, the springs must achieve a balance between the front and rear of the vehicle.
The next upgrade would be roll bars, which are helpful for cornering, but don't have an effect when going in a straight line. Again here, there are several combinations. Some call for a larger rear bar only, some call for no rear bar, some call for a larger front bar. Bottom line, the purpose is to maintain a balance, working with the springs. (In this case, the car has the stock roll bars, and of course, these have no effect on straight line driving...)
Last, small adjustments to the tire pressure do make a difference, but these are refinements. The stock set-up for the '84 calls for the front tires to have 2 lbs. less pressure than the rear tires. Doing the opposite does change the handling of the car, but it is hard to imagine that it can cause the rear wheels to spin. Whatever is wrong, is only accentuated when there is less friction, as in the case of rain. It is hard to imagine that a modern street tire that is 9 months old would have issues in the rain.
If it were me, I'd see if I could find a friend with a similar Spider and then I'd drive both cars. Then, I'd swap the wheels between the cars. If this doesn't change anything, then the problem is likely in the balance between the front and rear springs.
Best regards,