I could fix it manually with

--8<---------------cut here---------------start------------->8---
env LD_LIBRARY_PATH="/gnu/store/74i7r7qp1km0gw1i22fnq3szbgc9mpdx-sqlite-with-column-metadata-3.26.0/lib:$LD_LIBRARY_PATH" /gnu/store/2frbprf0qrqrk9v5wy0xfzkvmqk27wfk-ring-client-gnome-20190108.1.8659b2c/bin/gnome-ring
--8<---------------cut here---------------end--------------->8---

Something weird: the original error message

--8<---------------cut here---------------start------------->8---
/gnu/store/6q1ysbyki4v1zidbcndvyrchm3jncs58-ring-client-gnome-20190108.1.8659b2c/bin/gnome-ring:
symbol lookup error:
/gnu/store/zng0ix6b6icm8f8r6cqr09ykiz6rgrpg-qtbase-5.11.2/lib/qt5/plugins/sqldrivers/libqsqlite.so:
undefined symbol: sqlite3_column_table_name16
--8<---------------cut here---------------end--------------->8---

suggests that our build of qtbase lacks the symbol.  A "strings libqsqlite.so"
reveals that the symbol is there.  Besdies:

--8<---------------cut here---------------start------------->8---
> ldd /gnu/store/zng0ix6b6icm8f8r6cqr09ykiz6rgrpg-qtbase-5.11.2/lib/qt5/plugins/sqldrivers/libqsqlite.so
...
	libsqlite3.so.0 => /gnu/store/74i7r7qp1km0gw1i22fnq3szbgc9mpdx-sqlite-with-column-metadata-3.26.0/lib/libsqlite3.so.0 (0x00007ff81f146000)
--8<---------------cut here---------------end--------------->8---

So libqsqlite.so is linked against the right libsqlite3.so.0.  
This makes me wonder how the LD_LIBRARY_PATH trick above could work at all.

-- 
Pierre Neidhardt
https://ambrevar.xyz/