  We develop a modified CCZ4 formulation of the Einstein equations in $d+1$
spacetime dimensions for general relativity plus a Gauss-Bonnet term, as well
as for the most general parity-invariant scalar-tensor theory of gravity up to
four derivatives. We demonstrate well-posedness for both theories and provide
full expressions for their implementation in numerical relativity codes. As a
proof of concept, we study the so-called ``stealth-scalarisation'' induced by
the spin of the remnant black hole after merger. As in previous studies using
alternative gauges we find that the scalarisation occurs too late after merger
to impact on the tensor waveform, unless the parameters are finely tuned.
Naively increasing the coupling to accelerate the growth of the scalar field
risks a breakdown of the effective field theory, and therefore well-posedness,
as the evolution is pushed into the strongly coupled regime. Observation of
such an effect would therefore rely on the detection of the scalar radiation
that is produced during scalarisation. This work provides a basis on which
further studies can be undertaken using codes that employ a moving-punctures
approach to managing singularities in the numerical domain. It is therefore an
important step forward in our ability to analyse modifications of general
relativity in gravitational wave observations.
