Bingo! It's Working
Brand new Fire HD 8 running 22.214.171.124
I followed the directions in this thread - getting the correct android versions that matched the Fire OS base is what finally did the trick for me. Google Play is up and I can download and install to the Fire HD.
At first it would not run the SmartThings app from Google Play. I downloaded Chrome browser and set it as the default browser on the Fire HD. I restarted and checked that Chrome was still the default browser. I expected it would have reverted to Silk but not so (thanks, Amazon). I started SmartThings and was able to login and can now control all of my Z-Wave devices from the Fire HD.
Not bad for a $50 tablet. w00t!
The following variants are working for me:
Google Account Manager 7.1.2 - com.google.android.gsf.login_7.1.2-25_minAPI23(nodpi)_apkmirror.com.apk
Google Services Framework 7.1.2 - com.google.android.gsf_7.1.2-25_minAPI25(nodpi)_apkmirror.com.apk
Google Play Services 14.3.66 - com.google.android.gms_14.3.66_(040400-213742215)-14366019_minAPI23(arm64-v8a,armeabi-v7a)(nodpi)_apkmirror.com.apk
Google Play Store 11.9.14-all - com.android.vending_11.9.14-all_0_PR_214884739-81191400_minAPI16(armeabi,armeabi-v7a,mips,mips64,x86,x86_64)(240,320,480dpi)_apkmir ror.com.apk