The answer is a big NO. I will show this with one (very naive) counter example.
Assume that we should use relativistic mass in $v=-\frac{GMm}{r}$. Let's say S is inertial observer who moves in $z$ direction with veloctiy $u$. Sun is at the rest at $(x,y,z)=(0,0,0)$ and Earth orbits the Sun in $xy$ plane with velocity $w$. Now, from S's perspective Earth and Sun move in $z$ direction with velocity $-u$ and based on our assumption we should use relativistic mass for Earth and Sun. The velocity of Earth in $xy$ plan is small enough compare to $-u$ so it's safe to neglet it for now. Suppose that Sun's invariant mass is $M$ and Earth's invariant mass is $m$. According to S, their relativistic mass will be $M'=\gamma_uM$ and $m'=\gamma_um$ respectively. 499
If we put them in $v$ we will see that $v'=-\gamma_u^2\frac{GMm}{r}$ which is a lot greater than $v$. The problem is, Earth's velocity in $xy$ plane will become slower by a $\gamma_u$ factor from S's point of view. You can see it from velocity addition formula. Now, $F'=-\nabla_rv'$ So $F'_g=\gamma_u^2\frac{GMm}{r^2}$.
Earth-Sun system is in equilibrium (that's centrifugal force is equal to gravitional force in Sun's frame hence Earth doesn't change its orbit, roughly), So in Sun's frame we have $F_g-F_c=\frac{GMm}{r^2}-\frac{mw^2}{r}=0$. According to first special relativity principle, Earth should not change its orbit in other inertial frames like S, that's $F'_g-F'_c=0$, but unfortunately it's not the case because $F'_g-F'_c=\gamma_u^2\frac{GMm}{r^2}-\frac{mw^2}{\gamma_ur}\ne 0$ clearly. Which shows that assuming relativstic mass is not a good idea.
<PERSON>: Actually, instead of velocity addition formula, one can check Earth's motion in $xy$ plane from S's point of view with time dilation formula, That's, Earth orbits the Sun every $365$day from Sun's point of view, So from S's perspective it should take $\gamma_u 365$day for earth to complete its rotation. That would mean Earth's angular velocity is decreased by a gamma factor. If you want to consider relativistic effect, you will need <PERSON> field equations. 766