How to visualize the real projective plane $\mathbb RP^2$ in three dimensions, if possible?

It depends on what you mean by "visualize".

The standard mathematical interpretation of "visualizable in three dimensions" is that there exists an embedding in three dimensions. Such embedding cannot exist: a compact non-orientable $m-$manifold cannot embed in $\mathbb{R}^{m+1}$. There are several proofs for this: via Alexander duality, intersection number etc, and it is a non-trivial result.

Another interpretation can be that there exists an immersion in three dimensions, which allows self-intersections, for instance. Under this interpretation, it indeed is visualizable: the Boy's surface is an example.

But restricting to those concepts of visualizable is doing the enhancement of abstract geometrical thinking a disservice, imho. I am very content of saying that I visualize the projective plane by, say, visualizing $D^2$ with certain identifications on boundary, and this is arguably a two-dimensional visualization.