Get the AWD drivetrain swap. Also try reducing the front spring stiffness to about 440 lb/in and the rear stiffness to 390lb/in.
This, front wheel drive as a rule is really only viable to about 250hp/250ft-lbs. After that, as a rule, you have to move to AWD to keep a stable platform. Some cars like the new Focus RS can push beyond this in real life, but the suspension setups needed to make the car manageable are fairly exotic.
This is due to 2 primary reasons:
1. Torque steer. With a transverse mounted engine (which is what most fwd systems use), the drive shafts leading to the front wheels are uneven in length. This means that the wheels will tend to want spin at different rates, making the car want to go in a circle when under heavy acceleration. This can be corrected with a differential system, which is what most cars do. This is a very difficult thing to tune correctly, though, and even in modern cars there's occasionally some torque steer at high levels of torque. This can also be corrected with longitudinal mounted engine FWD systems, but this both limits the size of the engine and gets rid of most of the efficiency advantages of an fwd system. The main reason carmakers would use a longitudinal design is a forward mounted longitudinal engine is a very good platform for an awd version of a car, and in the cases that it is used the AWD version is usually the primary design and the fwd is a secondary, cheaper alternative for lower priced trim levels (Audi does this a lot).
2. Wheel lift-off. Weight shifts to the back of a car when accelerating. This is a problem for front wheel drive cars. First of all, it tends to greatly increase the likelihood of losing traction while accelerating. This makes it difficult to put power to effective use. In a rear wheel or all wheel drive car, when you accelerate quickly, the car shifts weight to the back, increasing the amount of traction you have on the rear wheels, and helping you maintain traction, thus letting you put more power down. With a front wheel drive car, the weight shifts to the back, and the front wheels come up. This means that you have less traction on the wheels that are providing the acceleration, giving you less effective power and increasing the likelihood of wheel spin.
Also, when turning, the inside wheels lose pressure on the ground, basically try to come up off the ground as the weight shifts to the outside of the car. In a rear wheel or all wheel drive car, this causes 2 effects: the first is that a loss in steering effeciency occurs at the front of the car. Rather than having 2 tires fully engaged to provide a change in direction, you only have 1. This makes the car harder to steer and thus turn slower. However, there's a counterbalancing effect going on at the rear wheels. There, the outside wheel is being pushed to the ground, while the inside wheel is lifting off of the ground (obviously it doesn't come off the ground completely in most cases, but the effect is the same). This means that the outside wheel is putting down more power than the inside wheel. For the same reason as why torque steer occurs this makes the car want to turn into the corner. The balance between effects 1 and 2 determines how much a car wants to understeer vs oversteer. Understeer being a complete loss of traction which means the car just keeps going in the same direction, oversteer being a situation where the car turns too fast and spins out. Basically, if you take a corner too fast and you end up hitting the wall head on, that's understeer. If you spin out and hit the wall with the back of the car, that's oversteer. Cars are most stable and turn best when neither understeer or oversteer dominate, this is called neutral or balanced handling.
Now, in a front wheel drive car, effect 1 happens as normal. However, when effect 2 happens, its not the outside rear wheel that takes up the slack but the outside front wheel. Because of the position of the outside front tire, oversteer isn't nearly as much of a factor (think of a lever where the axis of rotation is at the front, obviously a force towards the rear far away from the axis of rotation is going to cause more of a change in rotation than one at the front right next to it.) Also, a tire basically has a certain area with which it can interact with the road. In a hard cornering situation in a front wheel drive car, you have all the work of the car being done by one tire, rather than 2 in an all wheel or rear wheel car. This tends to cause the tire to lose traction more readilly. Both these factors together mean front wheel cars have little oversteer, and understeer dominates. This obviously becomes more of a problem the faster the car is going, and the more power it has (due to the combination of steering+acceleration being localized to only one wheel while cornering).
Putting an AWD system in the car will balance out the handling and make life much easier.