Not saying its a good idea but could it be done

Zero chance the Canucks give up their 1st, especially without protection. No team is giving up a shot at Bedard (except Bill Zito)

Arizon's already done a good enough job of putting themselves in the Bedard conversation on their own. If we assume that both teams pick where they currently stand (Arizona #4 & Vancouver #7), I don't think Arizona does it. Those two picks aren't going to be enough to jump into the top 2 and get the big name or the consolation prize - and now they're stuck with an aging JT Miller for 7 years at $8M per?

And, as always, it bears mentioning that players are going to be acutely aware of you trading a guy you just extended, before that extension kicks in, and more importantly, before that NMC you gave him kicks in. It's just a terrible look for management and should only be done in case of emergency (and I'd argue "acquiring Jakub Chychrun" doesn't qualify as one.)

Arizona does not want to be saddled with JT Miller given his current contract, and it seems like their price for Chychrun is way higher than that. Even though he's requested a trade, they're unlikely to trade him until they get the deal they want. Also, Miller makes Arizona better right now, which they absolutely do not want because as you pointed out they want Bedard.

Doesn't Chychrun also make them better right now? I imagine the exchange of players would not effect the standings.

Yes but the point is that if they are going to trade Chychrun theres no reason why they would target a player who will make them better as a central part of the trade.