Step down transformer when wattage is close
The transformer rating is 660VA. That means, that it can supply a load of 660VA continuously all the time.
Your appliance is fused with 3.15A fuse, which in practice means that continuous current is even less than that, let's say approx. 2.8A. That would be near 280VA continuous.In short bursts the load is increased to 700VA, but never more than 280VA over a period of time (mean value).
So your transformer is now over sized, you could be using 300VA.
As an example from one of my applications:

The heat sealing wire has approx 10kW power and it is supplied by a transformer of 2.4kVA. It works for 2s and it cools for 3min. Duty cycle approx. 1%

You can overload the transformer for various times, if you need just a small burst. The only limitation is, that the secondary voltage sags if you expect too much. Therefore a transformer of 100VA (1% of 10kVA) is not suitable, so there is still a large transformer which is completely cool as it does nothing most of the time.
EDIT:
Beside using smaller transformer, you could use an auto transformer 240/100 400VA it would have a weight of approx 1.7kg.