Tips for desoldering difficult to desolder mini transformers
It sounds as if one of your problems is the high temperature needed to melt the solder.  There is a commercial solution called "Chip Quik" ChipQuik that should help.

Basically, this is a very-low melt point solder that is added to the existing solder on the board.  The resulting mix of solder metals has a much lower melt point than regular solder and you can usually manipulate the plastic bobbin without melting the plastic while the solder is still molten.

You have to be careful to remove all of the contaminated solder before resoldering the both the removed part as well as the PC board but that is easy with any of the standard solder-removal techniques (solder wick, vacuum desoldering station, etc).

ChipQuik used to send out free samples, enough to remove a few chips.  The company was recently sold and I don't know if they still do that - but it's worth asking.

The free sample I received many years ago convinced me that the product was effective and I now keep it around on both my lab bench as well as our rework stations.