  Detecting stochastic background radiation of cosmological origin is an
exciting possibility for current and future gravitational-wave (GW) detectors.
However, distinguishing it from other stochastic processes, such as
instrumental noise and astrophysical backgrounds, is challenging. It is even
more delicate for the space-based GW observatory LISA since it cannot correlate
its observations with other detectors, unlike today's terrestrial network.
Nonetheless, with multiple measurements across the constellation and high
accuracy in the noise level, detection is still possible. In the context of GW
background detection, previous studies have assumed that instrumental noise has
a known, possibly parameterized, spectral shape. To make our analysis robust
against imperfect knowledge of the instrumental noise, we challenge this
crucial assumption and assume that the single-link interferometric noises have
an arbitrary and unknown spectrum. We investigate possible ways of separating
instrumental and GW contributions by using realistic LISA data simulations with
time-varying arms and second-generation time-delay interferometry. By fitting a
generic spline model to the interferometer noise and a power-law template to
the signal, we can detect GW stochastic backgrounds up to energy density levels
comparable with fixed-shape models. We also demonstrate that we can probe a
region of the GW background parameter space that today's detectors cannot
access.
