Is it possible to make a systemd unit wait until all its Conflicts= are stopped before trying to start

You can do this by adding the Conflicts= directive to the [Unit] section of your two unit files (Conflicts= in just one unit file works, too). From the man page for systemd.unit(5):

A space-separated list of unit names. Configures negative requirement dependencies. If a unit has a Conflicts= setting on another unit, starting the former will stop the latter and vice versa. Note that this setting is independent of and orthogonal to the After= and Before= ordering dependencies.

I found the answer: I needed to change the last line of the service file to:

WantedBy=postgresql.service

This way, whenever Postgres is started, my service is started too - but if my service fails, that doesn't stop Postgres.

Directives in the [Install] section only affect enabling and disabling of units. But it wasn't this simple when my service was already enabled:

# systemctl enable tabill.service
Failed to execute operation: Too many levels of symbolic links

The error message was misleading. Fixing it was simple:

# systemctl disable tabill.service
Removed symlink /etc/systemd/system/tabill.service.
Removed symlink /etc/systemd/system/multi-user.target.wants/tabill.service.

# systemctl enable tabill.service
Failed to execute operation: No such file or directory

# systemctl enable /srv/tabill/tabill.service
Created symlink from /etc/systemd/system/postgresql.service.wants/tabill.service to /srv/tabill/tabill.service.
Created symlink from /etc/systemd/system/tabill.service to /srv/tabill/tabill.service.

Now my service stops and starts whenever Postgres does. And naturally Postgres starts when the system boots.