Well, a failed headgasket would be constant white smoke (and mine was billowing white...), its better to do a CO2 check (kit is 65.00 on Amazon or I think you can rent one from an autoparts store) and I would really make sure you track down the short. Supras are adverse to fire... lol. I had a battery blow up due to faulty wiring and a short and it lifted my front wheels about 6" off the ground. Luckily the hood was closed when it overcharged/shorted.
Normally the signs of a bad headgasket are oil in the coolant and your coolant reservoir raising in level when running warm (can actually pump coolant out of there). Intermittent white smoke can be due to forged pistons depending on your engine. I could smell my bad headgaskets. Now... Im not a mechanic so what I say here is based on second hand info, but my pistons have a .04 tolerance due to forged pistons (yours may have forged, just judging from looking at it). If I idle for a while, I will end up with smoke due to oil slipping past those tolerances, but it will stop after I increase my RPMs. I am used to that on V8's - where high power builds will smoke a bit, but when I saw it first time on my 7M I kinda panicked. Do you know your engine specs on your car? It helps to have a background and know what you currently have.
We tend to think bad head gasket with 7M when its not always the issue, but with a newly purchased Supra I have seen that normally being the first problem to rear its ugly head quite often. (all but three of the Supras I have owned had head gasket issues due to negligence and bad repairs.
Remember, everything I say here is based on reading and experience, I am not a mechanic, so you cant take what I say as gospel truth, there are others here that are far more qualified to answer. (Just saying... cause I hate advising when I am not 100% sure)