Reshaping a plum/mirabelle

I have a Mirabelle de Nancy which is supposed to be a dwarf tree (max 8 foot).  It's near enough 20 foot tall and almost as wide, much bigger than the advertised size and is now over shadowing other cordon fruits nearby.

It's not only too big for its location but despite it flowering really well I can't cover it and pigeons strip every flower from it. (Oddly they don't touch my other plums or my greengage cordons.)

I need to cut it back hard to reduce the size and control the shape. Is this an option and just how hard can/should I cut it? It has a forked double trunk which would be fine with lower side shoots but would I kill it if I cut it right back to that degree?