The condition that $u(x-y)=g_1(x)f_1(y) + g_2(x)f_2(y) + f_3(y)$ immediately implies that $u(x)$ satisfies a kind of addition theorem and also linear recurrence relations with constant coefficients. Since there are at most $3$ linearly independent terms on the right side, one recursion is as follows: $u(x)=c_1u(x-1)+c_2u(x-2)+c_3u(x-3)\;$ for some constants $c_1,c_2,c_3.$ The theory of solutions to such recursions is well-known. Of course, you want $u(x)$ to satisfy condition (1) so that restricts the solutions. Aside from the quadratic solutions that you found, and which correspond to double root $1$, there are solutions $u(x)=-k_1(r^x+r^{-x})+k_2$ where $r\ne 0$ is real. The $r^x+r^{-x}$ can also be written using $\cosh(cx)$.

The condition that $u(x-y)=g_1(x)f_1(y) + g_2(x)f_2(y) + f_3(y)$ immediately implies that $u(x)$ satisfies a kind of addition theorem and also linear recurrence relations with constant coefficients. Since there are at most $3$ linearly independent terms on the right side, one recursion is as follows: $u(x)=c_1u(x-1)+c_2u(x-2)+c_3u(x-3)\;$ for some constants $c_1,c_2,c_3.$ The theory of solutions to such recursions is well-known. Of course, you want $u(x)$ to satisfy condition (1) so that restricts the solutions. Aside from the quadratic solutions that you found, and which correspond to double root of $1$, there are solutions $u(x)=-k_1(r^x+r^{-x})+k_2$ where $r\ne 0$ is real, and the $r^x+r^{-x}$ can also be written in the form $\cosh(cx).$