Deterministic context-free languages are closed under regular right-product
Here is an idea. You have to work out the details by your own.

Let $Q_R$ be the states of a DEA accepting $R$, and let $Q_L$ be the states of a deterministic PDA accepting $L$. You build now a new deterministic PDA $P$ that accepts $LR$ as follows:

The state space of $P$ is $Q'=Q_L\times \mathcal{P}(Q_R)$. If you have read $w$ and you are with $P$ in state $(s,T)$, this tells you the following: $L$ would be in state $s$ after reading $w$, and for every $t\in T$ there was a possibility that a prefix of $w$ was in $L$, and the remaining suffix of $w$ would have send the DEA for $R$ to state $t$. You accept if you are in a state $(s,T)$, that contains an accepting DEA-state in $T$. The transitions can be worked out straight-forward. Notice that it might be helpful to modify the PDA for $L$ first, such that it has a total transition function. You can achieve this easily by adding a non-accepting dummy states, that collects all missing transitions.