  We simulate a radial velocity (RV) follow-up of the TRAPPIST-1 system, a
faithful representative of M dwarfs hosting transiting Earth-sized exoplanets
to be observed with SPIRou in the months to come. We generate a RV curve
containing the signature of the 7 transiting TRAPPIST-1 planets and a realistic
stellar activity curve statistically compatible with the light curve obtained
with the K2 mission. We find a +/-5 m/s stellar activity signal comparable in
amplitude with the planet signal. Using various sampling schemes and white
noise levels, we create time-series from which we estimate the masses of the 7
planets. We find that the precision on the mass estimates is dominated by (i)
the white noise level for planets c, f and e and (ii) the stellar actvitity
signal for planets b, d and h. In particular, the activity signal completely
outshines the RV signatures of planets d and h that remain undetected
regardless of the RV curve sampling and level of white noise in the dataset. We
find that a RV follow-up of TRAPPIST-1 using SPIRou alone would likely result
in an insufficient coverage of the rapidly evolving activity signal of the
star, especially with bright-time observations only, making statistical methods
such as Gaussian Process Regression hardly capable of firmly detecting planet f
and accurately recovering the mass of planet g. In contrast, we show that using
bi-site observations with good longitudinal complementary would allow for a
more accurate filtering of the stellar activity RV signal.
