I found a solution, documenting for others…
wget https://sqlite.org/2021/sqlite-autoconf-3360000.tar.gz
tar -xvf sqlite-autoconf-3360000.tar.gz
cd sqlite-autoconf-3360000
./configure
make
sudo make install
hass still complained. It turned out to be where the libraries were installed.
The old libraries were in /usr/lib/arm-linux-gnueabihf/, whereas the new ones were in /usr/local/lib.
At that point I took the easy option (hass is the only application running, so no risk to other apps)…
sudo cp /usr/local/lib/*sql* /usr/lib/arm-linux-gnueabihf/
sudo chmod a+x /usr/lib/arm-linux-gnueabihf/*sql*
hass then restated without the warning.