  We consider a near-critical binary mixture with addition of antagonistic salt
confined between weakly charged and selective surfaces. A mesoscopic functional
for this system is developed from a microscopic description by a systematic
coarse-graining procedure. The functional reduces to the Landau-Brazovskii
functional for amphiphilic systems for sufficiently large ratio between the
correlation length in the critical binary mixture and the screening length. Our
theoretical result agrees with the experimental observation [Sadakane et.al. J.
Chem. Phys. {\bf 139}, 234905 (2013)] that the antagonistic salt and surfactant
both lead to a similar mesoscopic structure. For very small salt concentration
$\rho_{ion}$ the Casimir potential is the same as in a presence of inorganic
salt. For larger $\rho_{ion}$ the Casimir potential takes a minimum followed by
a maximum for separations of order of tens of nanometers, and exhibits an
oscillatory decay very close to the critical point. For separations of tens of
nanometers the potential between surfaces with a linear size of hundreds of
nanometers can be of order of $k_BT$. We have verified that in the
experimentally studied samples [Sadakane et.al. J. Chem. Phys. {\bf 139},
234905 (2013), Leys et.al. Soft Matter {\bf 9}, 9326 (2013)] the decay length
is too small compared to the period of oscillations of the Casimir potential,
but the oscillatory force could be observed closer to the critical point.
