According to Wald's GR [...] e.g. in a case where there are several massive bodies in relative motion--there exists no natural set of curves whose comparison with geodesics could be used to define gravitational force.
Why not?
Why not, indeed.
What could be more "natural" than to make the required comparison ("with geodesics") for each participant (and its corresponding "curve" or trajectory) separately; thereby defining "gravitational force" (or more inclusively: "gravito-inertial force") in general, and evaluating it case by case, separately for each participant and each trial (and without concern for any possibility of "defining a preferred set of background observers").
Specificly, given (a section of) the trajectory of participant $A$ as an ordered set of coincidence events $\{ \varepsilon_{A O} ..., \varepsilon_{A Q} ..., \varepsilon_{A X} \} $ in which $A$ had taken part (having met and passed participants $O$, $Q$ and $W$, among others, in this order),
and given the ("geodesic"-based) values of interval ratios between pairs of the corresponding events in which $A$ had taken part in this trial,
i.e. the real number values of ratios
$$\frac{s^2[~\varepsilon_{A P}, \varepsilon_{A Q}~]}{s^2[~\varepsilon_{A O}, \varepsilon_{A X}~]}$$
for all pairs of participants (such as for instance $P$ and $Q$, and also including $O$ and $X$) whom $A$ had met in the course of the trial,
then the magnitude $|~\mathbf a_A[~Q~]~|$ of $A$'s acceleration at the (event of) the meeting and passing of participant $Q$ can thereby be expressed as
$$ \begin{array}{ll} |~\mathbf a_A[~Q~]~| := \frac{c}{\sqrt{\stackrel{~}{|~s^2[~\varepsilon_{A O}, \varepsilon_{A X}~]~|}}} \times {\text{Limit}}_{ \large{\left\{\frac{s^2[~\varepsilon_{A P}, \varepsilon_{A W}~]}{s^2[~\varepsilon_{A O}, \varepsilon_{A X}~]} \rightarrow 0 \right\} }} &~ \cr \scriptsize{ \left[ ~~ \sqrt{ \stackrel{~}{\frac{s^2[~\varepsilon_{A O}, \varepsilon_{A X}~]}{s^2[~\varepsilon_{A P}, \varepsilon_{A W}~]} }} \times \sqrt{
\eqalign{
\stackrel{~}{
\left(
\frac{s^2[~\varepsilon_{A P}, \varepsilon_{A W}~]}{s^2[~\varepsilon_{A P}, \varepsilon_{A Q}~]}
\right)}~
\left(
\frac{s^2[~\varepsilon_{A P}, \varepsilon_{A W}~]}{s^2[~\varepsilon_{A Q}, \varepsilon_{A W}~]}
\right)~ +
\left(
\frac{s^2[~\varepsilon_{A P}, \varepsilon_{A Q}~]}{s^2[~\varepsilon_{A Q}, \varepsilon_{A W}~]}
\right)~ +
\left(
\frac{s^2[~\varepsilon_{A Q}, \varepsilon_{A W}~]}{s^2[~\varepsilon_{A P}, \varepsilon_{A Q}~]}
\right)~ \\ - 2
\left(
\frac{s^2[~\varepsilon_{A P}, \varepsilon_{A W}~]}{s^2[~\varepsilon_{A P}, \varepsilon_{A Q}~]}
\right)~ - 2
\left(
\frac{s^2[~\varepsilon_{A P}, \varepsilon_{A W}~]}{s^2[~\varepsilon_{A Q}, \varepsilon_{A W}~]}
\right)~ - 2 } } ~~~ \right] }; \end{array} $$
and the average direction of $A$'s acceleration throughout a trial from event $\varepsilon_{A O}$ until event $\varepsilon_{A X}$ would be expressed in terms of families of suitable "geodesic" participants; namely as "towards" any one participant $B$ (if there exists one), for each pair of events in which $B$ took part as well, such as $\varepsilon_{A B P} \equiv \varepsilon_{A P}$ and $\varepsilon_{A B W} \equiv \varepsilon_{A W}$, for which
$$ \sqrt{ \frac{s^2[~\varepsilon_{A B P}, \varepsilon_{B Y}~]}{s^2[~\varepsilon_{A B P}, \varepsilon_{A B W}~]} } + \sqrt{ \frac{s^2[~\varepsilon_{B Y}, \varepsilon_{A B W}~]}{s^2[~\varepsilon_{A B P}, \varepsilon_{A B W}~]} } = 1, $$
and the instantaneous direction of $A$'s acceleration at the (event of) the meeting and passing of participant $Q$ would be expressed by the partial ordering of such families, wrt. a partial ordering of trials which all include event $\varepsilon_{A Q}$.
The corresponding "gravito-inertial force applicable to" participant $A$, in the trial under consideration, at the event $\varepsilon_{A Q}$ of having met and passed participant $Q$, would be "in the opposite direction", of magnitude
$$m_A~|~\mathbf a_A[~Q~]~|,$$
where $m_A$ is "the mass" of participant $A$.
Quite separately from these considerations we can of course distinguish
Case 1: [...] in the vicinity of the Earth [...] a preferred set of background observers [...] remain stationary
... i.e. one set (or even several different sets) of observers/participants who remained chronometrically rigid to each other, and
Case 2: [...] several massive bodies in relative motion
where no such "rigid sets" of participants may exist.
