Dac that has tone and timing?

Lampizator.  I've owned, listened to, and interviewed many DAC owners and reviewers in the past few years, always eager to find something new or improved.  The Lampizators are still brilliant. I own an Atlantic 3, but the highly popular Baltic would come close to your budget.  Newer Lampi models have significant improvements that keep them relevant.  Have not listened to one or two of the other highly regarded, such as the T+A 200 or the Gustard R26.  I would think any of these would be solid end-game DACs. Some do fall outside your budget.

I really like my Bryston DAC3

Heard no real difference between my RME ADI-2 Pro, which I tried to "upgrade" to T+A 200 DAC + PreAmp. Both are DAC+preamp, which is necessary for me. People’s mileage may vary. My speakers in question are Dynaudio Focus 20 (active), and the source is always digital - Limetree Lindemann Bridge with LPS.

RME is an intensely resolving and "rich" DAC. It doesn't add or subtract anything. 

A worthy search! I heartily recommend PS Audio Directstream DAC Mk II. $8K, but worth it.

Of the affordable new DACs in your price range I chose the Wyred4Sound 10th Anniversary DAC. I find it tonally dense without sacrificing high end extension, true timbre, excellent macro and micro dynamics, pinpoint imaging, and very good sound stage. Editors in a number of journals much liked it.  All say analogue like whatever that means to them. I do not know why it  does not get mentioned often on this forum.  It is probably surpassed by the Denafrips Terminator higher end models from what I read,  but I never auditioned them in my own systems. Another choice I have heard in a friends system is the Bricasti M1 that should be around your price range pre-owned.  You cannot go wrong with any of the recommendations from others in this thread.  Let your ear’s make the choice.