Why can't we reduce register automata to symbolic automata?
The basic difference is that register automata have a memory while symbolic finite automata do not. Over an infinite alphabet this gives RA a power that can't be replicated in SFAs. Symbolic automata can only recognize unary predicates, and other relations are much harder.

For a simple example, consider the language $L=\{xx\mid x\in\Sigma\}$. That is, the set of all strings of length 2 that repeat a character. There is an easy RA to achieve this: state 0 reads x and stores it in register 1, going to state 1 (i.e. $q_0\to (q_1,0,\mathsf{right})$), and state 1 goes to the accept state 2 if the current value matches (that is $(q_1,0)\to (q_2,\mathsf{stay})$, and $F=\{q_2\}$) and rejects otherwise.

Suppose that a SFA accepts this language, where $\Sigma$ is infinite. Because for all $x\in\Sigma$, $xx\in L$, every character must initially be permitted. That is, for all $x\in\Sigma$ there is a state $q_x$ such that $(q_0,\phi,q_x)\in\Delta$ and $\phi(x)$ is true. Since $\Delta$ (and $Q$) are finite, there are clearly a finite number of such states $q_x$. But they are being tasked with distinguishing an infinite number of possible outcomes, depending on which $x$ was read. More precisely, by the pigeonhole principle there must exist $x\ne y$ such that $q_x=q_y$. Then the SFA must accept the string $x$ starting from $q_x$, since $xx\in L$, but that means it also accepts the string $x$ starting from $q_y$, so $yx\in L$ as well and hence the SFA doesn't correctly recognize $L$.