Since you have it open, I would disconnect one wire/side of the secondary of the transformer and measure the output Voltage with it out of circuit. It says it should be 17 Volts with a 16 VA (Watt) rating so it has about a 1 Amp rating. 125 VAC vs. 100 VAC is a 5/4 ratio so I am guessing the secondary Voltage will be around 17 V x 5 / 4 = 21.25 V. And 21.25 V - 17 V = 4.25 V so you need to drop that Voltage to bring it down to 17 Volts.
1N4001 through 1N4007 rectifier diodes are all inexpensive and rated for 1 Amp. They mostly differ by their reverse Voltage and the lowest Vr is 50 Volts so you can choose any of them. The actual current draw of your device is not known and probably varies so start with a nominal forward Voltage drop of 0.75 Volts. Use 4.25 Volts or your actual drop from your actual measurement: 4.25V / 0.75V = 5.6. So try six diodes in series. You need conduction in BOTH directions so six in series going one way with six more also in series, going the other way in parallel.
That should drop your transformer Voltage to the 17 Volts that the rest of the circuit was designed for. The diodes can stand the current and Voltages so there should be no problems there. And if you provide even a little of air around the diodes, they should be able to dissipate any heat generated. The transformer will be delivering the current it is rated for so it should be OK. Any extra Watts should be dissipated by the diodes, not the transformer so probably also OK.
Hook it all up and measure the actual AC Voltage past the diodes. Adjust the number of diodes as needed to get that 17 VAC number.
Those diodes are around $0.10 each when purchased in 10s so just a bit over $1 US for the mod. Diodes, even a dozen or more of them, are a lot less expensive than a new transformer.