Best Integrated amp for under 4k

Rega Mira is a fine integrated amp hard to beat, just crummy input jacks (one of mine broke from a cardas interconnect)

The MCINTOSH 6900 OR SOMETHING BY ACCUPHASE.

One of the best, if not THE best, integrated amps on the market today is the Musical Fidelity Tri-Vista 300, which is reviewed in the most recent issue of TAS. The Tri-Vista 300 has a superb preamp section, and the power amp is rated at 350 wpc. Unfortunately, it sells for $6000. If you have the time and patience to try to find a used model or a demo, it's worth considering.

Eyal S8
I have compared the Belles 250i to good number of amps most of which cost 2 or 3 times more in price. All were well known brands and they all were easily outperformed.The Belles sounds sweet,open,transparent and spacious. It's seperated power supply is a feature you normally find in amps in the mega buck range. The preamp section has a tube stage only 1 or 2 I think. I do not own this product but if I was in the market this is the one I would buy. The price retail is $3000 and the power is 125/125. Oh yes, it looks great.

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.