opinions on best neutral sounding amp with DAC under $4K

I don't know if this would bother you or not. The Hegel 160 is made in China.

When I first saw your post title, the piece that immediately came to mind was the Hegel H160. Since you're already thinking about that one, you're on the right track. An excellent integrated with a pile of awards. Disclosure: we are a Hegel dealer

All my Apple devices are built and China and they perform superbly! So it doesn’t bother me, it is the quality no origin of manufacture.

I borrowed the 160 from my local dealer for a few days.  Sadly I was unimpressed.  Tried a few different power cords etc., and could not get it to sound good.   Decidedly mid-fi in my view despite the 'professional' reviews.

I say sadly, because the choices at that price point are very limited.  Just recently sold the Naim v1/nap 100 combo - about the same price.  Again not impressed.  Have finally settled on a McIntosh D150 digital pre, which is excellent.  Pick a power amp to suit your tastes.

Nice speakers would love to hear them.

I had great listening sessions with the hegel h160 with spendors, harbeths tannoys and kef. Did great on all three speakers. I really like the hegel sound. My hesitation on the 160 is the apple play features - seem likely to be dated too soon for such an investment.

Another idea is to see if you can find a demo or used h300 which has more power and a better dac. Since the h360 is out the 300 is bound to start to be discounted in price. I have a h200 and love it but it has no dac.

Could look at the vinnie rossie modular amo too but may be slightly above your budget but its great.

Haven't heard parasound halo but there are many fans out there. 

Not too many other options i can think of. 

Keep us posted on how it goes.