One of the problems with hot-wire cutting is that the wire expands with temperature and a tensioning system is required. You _could_ use this to your advantage and control wire temperature by controlling wire length. This in turn could allow very simple control from your AC transformer through a 'phase-angle firing module'. These are glorified dimmer controllers with a DC control input.

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Your challenge with this approach is to come up with the linear transducer and some means of calibration. Setup would be by trial and error.