Could a modification of Krom's proof system be used to solve 3-SAT in polynomial time?
Any proof in KPS (with whatever rules) can be rephrased as a sequence of lines, each of which is either a clause or follows from preceding lines according to one of finitely many inference rules. Additionally, the lines come from one of finitely many "patterns" (formulas with placeholders to variables), and so all proofs are polynomially bounded.

Suppose that the maximum depth of a pattern is $d$. Then your proof is also a proof in depth $d$ Frege. It is known that the pigeonhole principle needs $2^{\Omega(n^{1/5^d})}$ lines to refute in depth $d$ Frege. This implies that KPS systems cannot be complete for 3SAT, since all proofs in KPS are polynomially bounded.