Assuming $X_t$ and $Y_t$ are diffusion processes and $Z_t=X_t Y_t$, then
$$
dZ_t/Z_t = dX_t/X_t+dY_t/Y_t + ... dt
$$
So if the diffusive parts of $X_t$, $Y_t$ and $Z_t$ are SABR, then
$$
\sigma^Z_t Z_t^{\beta^Z-1} dW^Z_t + ... dt = \sigma^X_t X_t^{\beta^X-1} dW^X_t + \sigma^Y_t Y_t^{\beta^Y-1} dW^Y_t + ... dt
$$
and a first necessary condition is $\beta^X = \beta^Y = \beta^Z = 1$.
Next, you have $\sigma^Z_t=\sqrt{(\sigma^X_t)^2 + (\sigma^Y_t)^2 + 2 \rho_{XY}\sigma^X_t \sigma^Y_t}$ and
$$
d\sigma^Z_t/\sigma^Z_t = ((\sigma^X_t)^2 d\sigma^X_t/\sigma^X_t + (\sigma^Y_t)^2 d\sigma^Y_t/\sigma^Y_t+\rho_{XY}\sigma^X_t \sigma^Y_t(d\sigma^X_t/\sigma^X_t+d\sigma^Y_t/\sigma^Y_t))/(\sigma^Z_t)^2 \\
+ ... dt
$$
and another necessary condition is that $d\sigma^X_t/\sigma^X_t = d\sigma^Y_t/\sigma^Y_t$, which in turns implies that if $X_t$, $Y_t$ and $Z_t$ are SABR, then $\alpha^X=\alpha^Y=\alpha^Z$, $\rho_{XY}$ is constant, and $\sigma^X_t/\sigma^X_0 = \sigma^Y_t/\sigma^Y_0 = \sigma^Z_t/\sigma^Z_0$ (all three stochastic volatilities are driven by the same brownian motion and have the same volvol).
The SABR parameters $\rho^Z$ is then computed as
$$
\rho^Z = (\sigma^X_0 \rho^X + \sigma^Y_0 \rho^Y)/\sqrt{(\sigma^X_0)^2 + (\sigma^Y_0)^2 + 2 \rho_{XY}\sigma^X_0 \sigma^Y_0}
$$
and an additional necessary condition is that $|\rho^Z| \leq 1$.
It's easily checked that put together, these necessary conditions are also sufficient, but it's all rather restrictive.
That being said, SABR is usually not used as a model but rather as a sparse parameterization of the implied volatility smile, so there is nothing wrong with fitting distinct SABR to each of the currency pair in the triangle, although the vanna-volga parameterization might be more popular for FX smiles.