Generally speaking, if you have two or three sources of noise, you are still going to be much better off pricing American options on a lattice than via LSMC. Too often, LSMC becomes the refuge of academics lacking patience to learn proper lattice techniques.
Now, you can frequently reduce the difficulty of pricing American options by considering the american exercise premium $P$, defined as the difference in value between an american-exercise option and its european-exercise equivalent
$$
P = A - E
$$
If you have some complicated stochastic model, but enjoy a technique $f(\cdot)$ for pricing european-exercise options
$$
\tilde{E} = f(x_E;\vec\mu)
$$
and you can define some much simpler model $g(\cdot)$ that is good enough for estimating the premium
$$
\tilde{P} \approx g(x_A; \vec\nu) - g(x_E; \vec\nu)
$$
then your american option price can be estimated as
$$
\tilde{A} \approx \tilde{E} + \tilde{P}
$$
If the american exercise premium is large then relative error in $\tilde{P}$ will be important and this trick will not work as well.
Also, if exercise probability is large, or exercise is likely to happen long before the option tenor, then the trick will fail, since we have introduced a dependency on $\vec\mu$ at (european) timescales well past the relevant timescales for the actual american option.