Ok, I found the answer myself, so I'll post it here. It turned out that the results for multipliers should be transferred not from $L^p(\mathbb{T})$ but from $L^p(\mathbb{Z})$. That is, the boundedness of operators that figure in the question follows from the boundedness of the following operator on $\ell^p(\mathbb{Z})$: we take a sequence in $\ell^p(\mathbb{Z})$, consider its Fourier transform (which is a function on $\mathbb{T}$), multiply it by characteristic function of an arc and take the Fourier coefficients of the resulting function. The boundedness of such operator on $l^p(\mathbb{Z})$ is of course known --- it can be found in the book by Edwards and Gaudry "Littlewood--Paley and multiplier theory" (1977).
As for this transference, it is known, too (and not very difficult): it is written in an article "Transference methods in analysis" by Coifman and Weiss (in a more general context; to be more specific, Theorem 3.15 and Corollary 3.16 can be applied here).