I find Cartesian coordinates let me easily solve a lot of navigation
problems on a sphere. I use them even for motion constrained to the surface
(so one of the coordinates is always redundant).
For notational convenience, let's write $r$ for radius, $\theta$ for latitude,
and $\varphi$ for longitude.
We want a new basis consisting of three orthogonal unit vectors,
which we'll call $\hat r$, $\hat \theta$, and $\hat \varphi$,
oriented according to the "up" direction at the point with
spherical coordinates $(r, \theta, \varphi)$
and Cartesian coordinates $(x,y,z)$.
The unit vector "straight up", $\hat r$, is the same as the unit vector from
the origin in the direction to your satellite's location,
$$
\hat r =
\begin{pmatrix}
\sin\theta \, \cos\varphi \\ \sin\theta \, \sin\varphi \\ \cos\theta
\end{pmatrix}.
$$
The unit vector $\hat\varphi$ points "due east". The satellite is
over the meridian at longitude $\varphi$; vector $\hat\varphi$ is perpendicular to the plane of that meridian and
is equal to a unit vector from the center of the earth
to the point on the equator at longitude $\varphi + \frac\pi2$, so
$$
\hat \varphi
= \begin{pmatrix} -\sin\varphi \\ \cos\varphi \\ 0 \end{pmatrix}.
$$
The third vector, $\hat\theta$, is the "northward" vector.
(When working in a coordinate system with $\theta = 0$ at the pole,
I think it may be more usual to define $\hat \theta$ in the exact
opposite direction, that is, pointing due "south",
in the direction of increasing $\theta$, but since it's also more common
to measure heading relative to due north than relative to due south,
a northward vector is a bit more convenient here.)
The northward vector $\hat\theta$
is the cross-product $\hat r \times \hat \varphi$
and can be immediately computed from the other two vectors using that formula.
Another way to find the coordinates of $\hat\theta$ is to consider
it as a vector an angle $\frac\pi2$ "northward" from $\hat r$.
That is, it is parallel to a vector pointing from the center of the earth
to a point at longitude $\varphi$ and latitude $\theta - \frac\pi2$.
When $\theta \geq \frac\pi2$ it is easy to see where this point is, but we
can also plot that point when $\theta < \frac\pi2$;
in the latter case it is the same as the point at
longitude $\varphi+\pi$, latitude $\frac\pi2 - \theta$.
In any case, the unit vector's coordinates are
$$
\hat \theta =
\begin{pmatrix}
-\cos\theta \, \cos\varphi \\ -\cos\theta \, \sin\varphi \\ \sin\theta
\end{pmatrix}
$$
(the same thing you wrote as a vector pointing due north, but in slightly more compact notation).
A much more succinct and elegant derivation of $\hat r$, $\hat\varphi$, and
$\hat\theta$ is in this answer to a similar question.
(I discovered this some time after initially writing up this answer,
and decided to keep the long version here in the hope that it may help
some people visualize the geometry.)
If you have the satellite's position in Cartesian coordinates, you can
find $\hat r$, $\hat \theta$, and $\hat \varphi$ as follows.
Using the fact that
$$
\begin{pmatrix} x \\ y \\ z \end{pmatrix} =
\begin{pmatrix}
r \sin\theta \, \cos\varphi \\ r \sin\theta \, \sin\varphi \\ r \cos\theta
\end{pmatrix},
$$
if $x^2 + y^2 \neq 0$ compute
\begin{align}
r &= \sqrt{x^2 + y^2 + z^2} \\
\hat r &=
\begin{pmatrix} x/r \\ y/r \\ z/r \end{pmatrix} \\
\rho &= r \sin\theta = \sqrt{x^2 + y^2} \\
\hat \varphi &= \begin{pmatrix} -y/\rho \\ x/\rho \\ 0 \end{pmatrix} \\
\hat \theta &=
\begin{pmatrix} -xz/(\rho r) \\ -yz/(\rho r) \\ \rho/r \end{pmatrix} \\
\end{align}
If $x^2 + y^2 = 0$ then the satellite is on the north-south axis and
"heading" has no meaning, so heading-pitch-magnitude will not give us
sufficient information to determine the direction of motion.
To convert a heading $\psi$ and pitch $\alpha$
(also known as a bearing or azimuth $\psi$ and elevation angle $\alpha$)
to a vector,
I would construct the vector using the three unit vectors
$\hat \theta$, $\hat \varphi$,
and $\hat r$ as components,
much as a vector was derived from elevation and azimuth in
this answer.
That is, letting $v$ be your velocity vector,
and assuming you know the magnitude $\|v\|$ of your velocity,
$$
v = \|v\| ((\cos\psi\,\cos\alpha)\hat \theta
+ (\sin\psi\,\cos\alpha)\hat \varphi + (\sin\alpha)\hat r).
$$
$\newcommand{\Brak}[1]{\langle #1 \rangle}\DeclareMathOperator{\proj}{proj}$If $A$ and $B \neq 0$ are vectors (in an arbitrary inner product space, with the inner product denoted by angle brackets), there exists a unique pair of vectors that are (respectively) parallel to $B$ and orthogonal to $B$, and whose sum is $A$. These vectors are, indeed, given by explicit formulas:
$$
\proj_{B}(A) = \frac{\Brak{A, B}}{\Brak{B, B}}\, B,\qquad
\proj_{B^{\perp}}(A) = A - \proj_{B}(A)
$$
The first is sometimes called the component of $A$ along $B$, and the second is the component of $A$ perpendicular/orthogonal to $B$.
The point is, the component of $A$ perpendicular to $B$ is unique (unles you have a definition that explicitly says otherwise) so "no", you need not/should not take both choices of sign.
Best Answer
Since $N$ and $V$ are penpendicular, we need
$$\langle n, v \rangle =0$$
Since $R$ and $V$ form an angle $\theta$.
$$\langle r, v \rangle = \cos \theta$$
We then solve the linear system to recover $v$.
Alternatively, use cross product to find unit vector $p$ that is perpendicular to both $n$ and $r$.
$$v = (\cos \theta) r \pm (\sin \theta) p$$