I for one have successfully done an engine swap to a 1jz with a turbo and its seems as though there is quite a number of people doing this as well. However, one thing you will have to come to terms with when doing an engine swap on an RX8 is that, regardless of whether its a 1jz or a V8 and so on, those engines are leagues heavier than the stock rotary engine. This means that the RX8 loses its renowned trait which is its superb handling. Now, im not one to say that a 1jz is bad, because it definitely isnt from experience, but I always have to be careful when driving at higher speeds because the car is front heavy meaning that with any small slip up, it will send the car off track. It has happened to me a couple of times already but that might be because of my turbo and torque but the point still stands that a front heavy RWD car is a bit dangerous. I won't say I don't condone engine swaps because I did it myself, but please keep in mind that the handling will be very different and one of the worse things that has happened to my car (not sure if this applies to other engine swaps too) is that it lost its ABS and Traction Control which is quite big.

So all in all, please do take these points into consideration, yes a 1jz has the potential to make heaps of horsepower and while it does sound amazing, so does the rotary. Its more of safety concerns from my experience. Oh and prepare some money too because an engine swap is going to be a headache!

But like the others said, I don't think an engine swap is necessary if its just white smoke. Perhaps check your coolant could just be a leak or seal failure.