Robert, The simple and inexpensive way to handle this problem is to set a double pole,single throw, normally open relay -120 volt coil voltage. This relay should have contacts that will have 20 amp inductive load rating. It should be available from any appliance supply house. The wiring is as follows: Connect a pair of leads to one pair of the contacts (these will need to have a 115 volt plug on the other end as they will be plugged into a wall outlet when the wiring is finished). Connect your enlarger lamp to the other pair of contacts on the relay. Connect the enlarger lamp output from your timer to the coil of the relay. The relay should be mounted in an enclosure and ground wires connect from the wall outlet to the enclosure chassis, from the enlarger lamp to the enclosure chassis.
This will shed the load from your enlarger timer and the enlarger timer will only need to carry the load of the relay coil (less then 150 watts). The relay contacts carry the load of the enlarger lamp. With a 20 amp inductive rating you will be able to handle enlarger lamp wattages of over 2000 watts.
If you have questions on the wiring, you may contact me off post. Good luck.