I looked at the listing for the LPL power supply at B&H (It looks like the same one as shown in the Adorama link). I believe that model is intended to power a 100 watt/12volt lamp on the LPL 670 and C6700 series dichroic lamp houses, so no, this probably won’t work on the Esco with its 250 watt/24 volt ELC lamp.
You could use the Omega Standard Power Supply #412-021 intended for the Omega D Dichro II Lamphouse. It is a 300 watt 120v to 22.5V step-down transformer intended to power the ELC lamp at a 6.25% voltage reduction (supposed to prolong the lamp life) in the Dichro II head. These can be found on eBay for reasonable prices. Wiring it to work with the Esco system in its original manner might be challenging.
You could certainly wire the lamp circuit of the head directly to that of the Omega power supply using an external timer. You’d also have to provide the cooling fan in the head with 120 volt current (I’m assuming that Esco provides a fan). You’d also need the Omega #404-841 Solid State Voltage Stabilizer or a 120 volt line stabilizer of sufficient power rating for color work. (Note: the Omega 75 voltage stabilizer is far too weak for a power supply of this size). The voltage stabilizer isn’t needed for B&W enlarging.
A simpler alternative is to look for an Omega D Dichro II head and power supply (and voltage stabilizer if you intend on color enlarging). These can be found on eBay for reasonable prices.
It would be great if you could find a technician that could repair your unit, but that might be tough to find.