The first rule of negotiation is to know where you draw the line, and not to make empty threats.
You've already said you don't actually want to move. Obviously the agency doesn't know this, but it puts you on the back foot immediately. So you're not really "negotiating", just "requesting".
I think all you can say is "we've been here 6 years, have never given any trouble. We have been impacted by the current situation, and would like to request a reduction in rent to €1,600 a month, which is still higher than other similar properties in the area."
Going the route of "We're intending to hand in our notice as we intend to rent a similar property for cheaper, we have not signed yet however" would obviously give this a lot more bite, but you don't want to go there and risk your current place, so you might be left just making a polite request.