 Solution

The identities $S_L^{†}{\bar{\sigma }}^{\mu }{S}_L={\Lambda }^{\mu }{}_{\nu }{\bar{\sigma }}^{\nu }$ and $S_L^T{\sigma }^2{S}_L={\sigma }^2$ show respectively that the kinetic term and both mass bilinears are Lorentz invariant. Infinitesimally, the first identity makes $u_L^{†}{\bar{\sigma }}^{\mu }{u}_L$ transform by the vector law stated in the question, while the second makes $u_L^T{\sigma }^2{u}_L$ a scalar.

Varying the anticommuting components of $u_L^{†}$ givesThe factor two from varying the antisymmetric quadratic form cancels the $1/2$ in the mass term. This is a Majorana mass term: under $u_L\mapsto e^{iq\alpha }{u}_L$ it carries charge $2q$, so a nonzero mass is compatible with an unbroken electric charge only for $q=0$. A charged massive fermion instead needs an independent Weyl field of opposite chirality to form a Dirac mass.