How can I show a linear languages are closed against concatenating with regular ones?
The piece of the puzzle is not to start $L_1$ and $L_2$ in parallel, but to first generate the regular suffix (strings in $L_2$), then (when the left-linear grammar finishes) continue with the linear prefix (strings in $L_1$).  

(added, deadline closed) Thus we assume we have grammar $G_i = (\Sigma,V_i,P_i,S_i)$ for language $L_i$, where $G_1$ is linear, and $G_2$ is left-linear, following your own suggestion, and generates the regular language "in reverse".
Given two strings $w_1$ and $w_2$ in the respective languages, having derivations $S_1 \Rightarrow^*_{G_1} w_1$ and $S_2 \Rightarrow^*_{G_2} B w' \Rightarrow_{G_2} w_2$, we glue these together as
$S_2 \Rightarrow^*_{G_2} Bw'\Rightarrow S_1 w_2 \Rightarrow^*_{G_1} w_1 w_2 $. Thus, first generate $w_2$, in the last step switching to the axiom $S_1$ of $G_1$, then generate $w_1$.

Technically this is achieved by requiring $V_1 \cap V_2 = \varnothing$, choosing all productions in $P_1$ and $P_2$ except that "final" productions $A\to a$ with $a\in\Sigma$ in $P_2$ are changed into $A\to S_1a$.