How hard is it to find the right transformer for led G4 bulbs

[ban-all-sheds]

Will this burn the light bulbs and melt too or can I just replace the lightbulbs?

It might be no good for LEDs at all

12V Effective - who knows what the frequency and peak-to-peak voltage is. Your LEDs might work, they might fail, they might work but for a shortened life.

Replace it with a proper LED supply.

What about the other black box with the writing Custom Transformers part number 4451 ...

Isolating transformer for the shaver socket.