Best Integrated amp for under 4k

Plinius 8200MK2. Not sure what your speakers but it bests with heavy loads such as Thiel or JM-lab. Used one is sold for less than $2k and will give you more room for speakers or even analogue since it has a phono.

What is your speakers?

I clearly see that with your speakers you will definitely benefit from separates or high-powered integrated which I'd pessimistically say not always has a true specs especially powerwise.
If you go with McCormack separates DNA225 and its passive pre TLC1 you'll be in budget and they're not spacey if that is the issue.

Never heard your speakers(well only Bach monitors from that line) but 8200 always seemed to me to bring plenty enough details and clarity + you never feel the lack of power.
In this case SIM I5 is much brighter than Plinius may work great with your speakers.