I would make sure the New offer is solid before you start renegoitaiting. If it's 100% happening let them know at that point and then ask for the no trade in deal. If you have given a large deposit unlikely they will see it's in their interest to discount the car but on the flip side if you are going to walk with no new deal they would be a poor dealer not to give your money back a day after the deal.
Sounds like you were happy with the deal and now potentially better off so positive, but always worth asking for more.
On the subject of deposits never leave more than you want. Porsche asked me for £2,000 BMW £1,000. The most I have ever given them is £500 and most times I offer them £250 and that's the lot. They all just try it on so play them at there own game.