ordering from China.....which Chinese Tube Integrated at 3k?

Black Ice (formerly Jolida) makes things in China but finishes, backs them up, and services them in the US. I owned a Jolida tube amp for years and it was a great sounding amp that worked perfectly. A bargain.

I had a LSA VT-70 (EL34 outputs) that I was very pleased with. They run about half the price you gave. They are Chinese made but you buy them from a US seller, Underwood Hifi.  LSA also has a larger unit, the VT-150 with KT88s that is still well under your max budget.

Consider this: https://www.musicalparadise.ca/store/index.php?route=product/product&path=66&product_id=55

Or any other of Garry Huang's remarkable products.

I would take the advice from @oddiofyl Quicksilver. 

I have a BlackIce tube DAC and love it. Also a bargain for how good it sounds. Jerred is great. US based small business.

I owned a Line Magnetic LM-88IA tube amp but bought it locally here in SoCal for $3K new, which I thought was very good for the money, so why buy from China? 

My understanding is that Line Magnetic manufactures their own products to quite high standards to compete well internationally, as opposed to a brand based in the US or anywhere else using a China-based contract manufacturer. Not that a contract manufacturer can’t make decent gear, but LM is directly responsible for engineering, manufacturing and quality control.