So long as you properly size your step-down transformer there is nothing wrong with running your existing 120V tools off the transformer if you are happy with your existing tools and do not mind that they will be running a but slower than you are perhaps accustomed to.
That said, as to what will make the most sense may depend in part upon how long you are likely to be living in Europe (1yr, 5rs, 20yrs, forever?) as that may influence your decision heavily. If you will only be living in Europe for a few years (or if you *may* only live there a few years) then a step-down transformer makes the most sense as transformers are not prohibitively expensive (on the secondary markets) -- provided, of course, that the cost of shipping heavy power tools overseas does not consume your potential savings (in which case the decision becomes easy regardless of duration, unless there is a strong emotional attachment).
However, please be aware that all of your power tools utilizing induction motors will be running 1/6th slower in Europe, versus the US, as they will be synching with a 50Hz waveform rather than the intended 60Hz (in other words, a nominal 3600 RPM motor will run at 3000 RPM) -- and that is something no step-down transformer can correct for (frequency correction can only be achieved with an inverter that can then synthesize a 60Hz AC source at whatever desired voltage or via a motor-alternator arrangement). Depending upon where you presently live in Japan, you may, or may not, have already experienced this slowdown -- some parts of Japan have 50Hz power while others use the same 60Hz power we are accustomed to in the US.
Enjoy your new adventure and be sure to keep us up to date on your saga.