These are general troubleshooting notes, I have no knowledge of the Beseler color heads.
Check all the wiring and crimp on connectors very carefully, also the bulb socket. If the copper wire in the crimp connectors is a brassy yellowish colour, there is a poor connection there and the wire is/was overheating. You will probably have to replace that piece of wire and the connector ends as it is rather difficult to remove the surface oxidation on the stranded wire to crimp or solder on a new connector.
High wattage halogen bulbs are almost a dead short until the filament warms up and the resistance rises to limit the current draw. Any poor connection in the circuit from the transformer to the bulb will limit the inrush current and extend the startup time of the bulb. This is also one of the causes of burnt power transformers. Burnt high current relay contacts between the transformer and the bulb will cause problems as well.
Clean the fan and oil the bearings. If there is an airflow sensor, that could delay the bulb start until the airflow is up to speed.
If there is an open frame relay, check for surface corrosion, darkrooms are rather hostile environments for electronics, if the contacts appear burnt, either polish them or replace the relay.
It doesn't take much to mess up a high current low voltage power feed to a 24 volt halogen bulb. Make sure any push on wire connectors are clean and tight. You can clean the blade terminals with a pencil eraser or a can of electronic contact cleaner and use a pair of pliers to slightly squeeze the push on fitting so it is a tight fit on the blade end.
Best of luck.
I'm not going to get into troubleshooting electronic control circuits, because if you know how, you probably would have been in there already.