what dacs are fast, lively, energetic?

I agree with those who are wondering about the pairing of “fast” and DAC. I have a Linn Klimax with Dynamik (2018 generation). Musical as hell, lots of air and separation. It is indeed capable of fast, but musical is the attribute that best describes it.
 

It is also insanely expensive new. I got mine on the used market and now will be sending it in for the Organik upgrade. The PS Audio DirectStream (original version) went to a backup system. No comparison, the Linn is night and day better (at 4x the cost). 

Squared 80 has a point and is squared like me. Lots of great DACs you can test drive before you buy.

+1 Chord DAC.

I never thought I would suggest looking at Topping, but here is a good review of the new Topping d90 III discrete - , discrete meaning it is not like their other DAC’s, it is not a R2R it is a single bit DAC.

Sandu, great reviewer, talks about exactly what the OP is asking for: speed and energy (jump to just after 8:00)

D90 III discrete

Could tide you over until you save up for your 5k budget (?) 

@mclinnguy He also did a review of the Gustard X-30 dac.  It's had at least three rave reviews, but still seems to be flying under the radar.  It's a long review.  Summary is that he really likes it.  :)

He doesn't describe it as fast, but to be honest, I'm not sure what that means. In my mind, that depends on the music.  

https://www.youtube.com/watch?v=VLkFoU3fu18