Personally I wouldn't plug the transformer into the extractor, for two reasons:
1. The extractor probably won't sense when the tools are being used. The most likely thing is that it will come on and stay on as soon as you plug in the transformer, although it might work fine - you'll have to try it. This is because there is a small current flowing through the transformer all the time, even when tools are not being used. The extractor will think that is a switched-on tool. But see (2) below...
2. It may damage your extractor when you turn the pair (extractor + transformer) on and off, or at the moment you plug the transformer into the extractor's tool-detecting socket (if the extractor already has power): The transformer produces a hefty voltage spike (technically "back EMF") when you turn it on and a bigger one when you turn it off. This can damage circuitry, and wear out switch contacts. Actual mains sockets (house or workshop wiring) are designed to cope with this (although they will wear too, it's very slight and will take a lot of use to do any noticable damage, and anyway a double mains socket plate is only a few quid).
My 110V transformer often trips the MCB on the workshop's consumer unit when I plug it in, but neither it nor the MCB are faulty - it's that pesky back EMF causing it.
Coincidentally, both my mitre and tracksaws are 110V. Apart from the extractor auto-switching, they're both fine to use. I just have to operate the extractor's switch or the remote instead - it's not too bad really.
You might do the other thing: get a set of those cheap remote mains switches from Maplins or the internet. I have some and they're fine switching my vacuum cleaner. They usually come with one remote and several switched sockets. It's the sockets that will wear out switching big motors, but you just use one and keep the others as spares. That's my preferred way to do it, and I have a decently long 110V extension cable I keep in the tracksaw's systainer, so I'm not lugging the transformer around when using it.
Hope that helps.
E.
