Here is a fairly crude first attempt.

First note that for any basis $B$, when $h=2$, we have \[0\leq r_{B,2}(n)-R_{B,2}(n)\leq 1\] for all n, as you note in your question. Hence assume $h\geq 3$. In this case, if $r'_{B,h}(n)=r_{B,h}(n)-R_{B,h}(n)$ counts the number of representations where some elements are identical, then \[ r'_{B,h}(n)=\sum_{m\in 2\cdot(B\cap[1,n])}r_{B,h-2}(n-m)\ll\lvert B_n\rvert r_{B,h-2}(n).\]

Hence if we can show that this upper bound is $o(r_{B,h}(n))$, then we have $R_{B,h}(n)\sim r_{B,h}(n)$ as required.

For example, note that this shows that distinct summands dominate in the case of Waring bases, where $\lvert B_n\rvert\approx n^{1/k}$ and $r_{B,h}(n)\approx n^{h/k-1}$.

Also note that it deals with e.g. the ternary Goldbach case, where $\lvert B_n\rvert\approx n/\log n$ and $r_{B,3}\gg n^2$.

This does not, however, deal with thin bases, where $r_{B,h}(n)\approx\log n$ and $\lvert B_n\rvert\approx n^{1/k}$. For such cases you may need to rely on probabilistic arguments, as in the proof of the Erdos-Tetalli theorem.

Here is a fairly crude first attempt.

First note that for any basis $B$, when $h=2$, we have

$$0\leq r_{B,2}(n)-R_{B,2}(n)\leq 1$$

for all n, as you note in your question. Hence assume $h\geq 3$. In this case, if $r'_{B,h}(n)=r_{B,h}(n)-R_{B,h}(n)$ counts the number of representations where some elements are identical, then

$$ r'_{B,h}(n)=\sum_{m\in 2\cdot(B\cap[1,n])}r_{B,h-2}(n-m)\ll\lvert B_n\rvert r_{B,h-2}(n).$$

Hence if we can show that this upper bound is $o(r_{B,h}(n))$, then we have $R_{B,h}(n)\sim r_{B,h}(n)$ as required.

For example, note that this shows that distinct summands dominate in the case of Waring bases, where $\lvert B_n\rvert\approx n^{1/k}$ and $r_{B,h}(n)\approx n^{h/k-1}$.

Also note that it deals with e.g. the ternary Goldbach case, where $\lvert B_n\rvert\approx n/\log n$ and $r_{B,3}\gg n^2$.

This does not, however, deal with thin bases, where $r_{B,h}(n)\approx\log n$ and $\lvert B_n\rvert\approx n^{1/k}$. For such cases you may need to rely on probabilistic arguments, as in the proof of the Erdos-Tetalli theorem.