Hi Pritch,
You are right in that this can be tricky to get the right equipment. For most LED lamps you will need to use a trailing edge dimmer such as the Artistic Licence Sundial already suggested. It is a 4 channel unit with a capacity of 250w per channel.
With regard to the lamps, you can find a list of tested options on the Sundial web page, but outside of this I always get a test one with a transformer if I'm installing a control system. Even then, you can run in to issues when multiples are connected to a circuit.
Although they are a bit more expensive, I recommend looking at the Soraa MR16 lamp as they have good compatibility. They also have some nice options in terms of accessories but not having seen your fixture I don't know if that's relevant. I think it would make sense to replace the lamp and the transformer at the same time. I don't think you would be able to mix halogens and LEDs on the same dimming line, so maybe you can do one circuit at a time and use the halogen bits you remove as spares for anything left?
Please let me know if you need any more info,
Maria