You don't need a transformer to go back down to 110 volts. A relay with a coil rated at 220 volts can be used to switch your enlarger on, connected to the 110 volt supply.
You could probably find a local electrician who could make this for you. However, it might be worth getting someone to look inside the timer to see if it can be converted to 110 volts. Often, equipment is fitted with a transformer with two primary windings, both 110 vlts. They are connected in parrallel for 110 volts and in series for 220 volts. You might be lucky!