I've figured out the cause of the problem and post the solution here, so it may be helpful to others.
tl;dr: Avoid version mismatch (due to automatic updates) between Google Play Services and Google Play Store apps in Fire HD profiles.
The two apps Google Play Services and Google Play Store seem to get automatically updated, no matter whether automatic updates are enabled inside of the Google Play Store app.
The original versions (as listed in the questions) manually installed from APKs were:
- Google Play Services: 11.5.09 (240-164803921)
- Google Play Store: 8.4.40.V-all  [PR] 176415973
Some time after installation I checked the version information for both apps by doing the following:
- Open the Settings app.
- Select the Apps and Games button below the Device group.
- Select the Manage All Apps button.
- Select the All tab.
- Scroll until Google Play Services and Google Play Store are visible in the list.
- Select the list entry for each app.
- Check and compare the version information (at the top).
By doing that I've noticed that the version of both apps has changed to the following:
- Google Play Services: 11.9.51 (234-177350961)
- Google Play Store: 8.5.39.W-all  [PR] 178322352
After I've downloaded the APKs for both versions from apkmirror.com (see links above) and installed them in the second profile the installation worked flawless.
I am now able to use Google Play apps in both the primary and secondary profile.
In conclusion the cause of the problem was a version mismatch between the Google Play Services and Google Play Store apps in the primary (newer, automatically updated versions) and secondary (tries to install older versions) Fire HD profile.