Pronunciation of Third Reich and Heinrich Himmler

Solution 1:

Reich has been in the English language for over two centuries, and thus has had time to assimilate to English pronunciation.

The OED ("Reich, n.") includes citations as early as the 18th and 19th centuries:

1762 P. Murdoch tr. A. F. Büsching New Syst. Geogr. IV. 4 The Empire [Ger. das deutsche Reich] is differently denominated as well by Germans themselves as by others. It is called the Reich [Ger. das Reich]..by way of eminence, as also the German Reich [Ger. das deutsche Reich].

1852 Times 6 July 6/4 It was the old court of appeal of the Reich, remarkable in its time, even among other courts, for its majestic slowness of procedure.

So it had two centuries to standardize its pronunciation to stay in line with other end-[ch] sounds. Greg Brooks categorizes these in his Dictionary of the British English Spelling System (available for free in PDF) as pronounced /k/ (irrespective of position), and examples include

aurochs

broch, loch, pibroch, Sassenach (when pronounced with /k/ rather than Scots /x/)

stomach, triptych, pentateuch, diptych, distich

cromlech (in some dialects it's /x/ rather than /k/)

as well as all the fun words where it doesn't appear at the end, like Christ, chianti, schooner, and masochist (p.283-4).

Meanwhile, German names (Brooks notes Schumacher; you note Heinrich) tend to keep phonemes closer to their language of origin, especially if the person is not native English-speaking. For example, Brooks says that Schumacher keeps the /x/ from German (p.286). For Heinrich, educated speakers may pronounce someone's name from Germany with /x/ (I've heard that pronunciation), but note that transition to /k/ has been ongoing for people who emigrated from Germany some time ago. For instance, here is how one person insists the [ch] in the name "Michael Heinrich Horse" be pronounced: /k/. (YouTube)