I am by no means an authority on this area but prompted by user /u/called2voyage I will refer the 1994 Astronomical Journal paper by Shahid-Saless (Colorado) and Yeomans (JPL) Relativistic Effects on the Motion of Asteroids and Comets.
To paraphrase their abstract: They study the predicted effects arising from relativistic perturbations on the motions of asteroids and comets and show that for a number of such objects, inclusion of relativistic contributions in the equations of motion give rise to significant improvements in the orbital solutions. They go on to argue that ignoring such corrections to the equations of motion gives incorrect solutions. They point out how the use of masses derived from relativistic ephemerides together with purely Newtonian equations of motion results in an inconsistent hybridized non-Newtonian, non-relativistic model.
With respect to the posted question:-
(a) are the existing observational data on any Asteroids or Comets sufficient (in terms of accuracy and extent of observations) to determine/estimate rates of non-Newtonian perihelion advance?
...the authors short-list 15 asteroids with the largest predicted relativistic perihelion precession rates (from a long-list of 156 commonly-studied objects). For example Icarus has the greatest rate (0.101 arcseconds/year) and had (in 1993) been observed over 43 years thereby producing a predicted cumulative relativistic precession over this period of 4.34 arcseconds. Observations referred to are a mix of optical and radar (no spacecraft telemetry) and are those used to produce the ephemerides model from which orbital elements are extracted.
The authors note that, assuming a (then) current observational accuracy of a few tenths of a second of arc, the relativistic contribution to precession should be detectable for Icarus and other asteroids near the top of the (ranked by precession-rate) list. As with the planets the majority of asteroid perihelion precession will be caused by the perturbing influences of other planets, but the authors do not present calculated values for these contributions).
(b) are any such determinations publically available?
...the authors selected six asteroids whose motions are significantly affected by general relativistic effects. They then calculated a set of orbital elements for each orbit using JPL's development ephemeris DE200 (with Earth and Moon perturbations treated separately). They did this in two ways, firstly with Newtonian equations of motion alone and secondly with the addition of non-Newtonian General Relativisitic equations of motion.
The authors indicate that the residuals (differences between observed and predicted positions at various times) are improved (i.e. reduced) by application of non-Newtonian corrections to the equations of motion. For example the RMS residual for Icarus is improved by 30%. Thus it is technically possible to calculate differences in the orbital behavior (such as perihelion advance) predicted by (i) the pure-Newtonian model and (ii) the Newtonian plus non-Newtonian model.
However the paper does not actually present explicit accounts of total observed perihelion precession or a breakdown into Newtonian and non-Newtonian components. In any case, due to complexity, such a breakdown could only apply over the period of observations which is limited (maximum 61 years in 1993) compared to the timescale (a few hundreds of years) over which (predominantly Newtonian) planetary-induced perturbations might be expected to average out.
However it would be possible (assuming there are no major perturbing encounters with planets or the asteroid belt) to use the provided asteroid orbital element data in a numerical orbit simulator program to roughly model the perihelion precession rate of an asteroid orbiting the Sun in the presence of the other planets using Newtonian equations of motion (i) without and (ii) with relativistic modifications calculated from $\frac{F}{m} = \frac{GM}{r^2} \left(1+\frac{3V_t^2}{c^2}\right) = \frac{GM}{r^2} \left(1+\frac{3GM.SLR}{c^2.r^2}\right)$ (see Walter-Goldstein-Schwartzchild Formulation ). That way the amount of asteroid precession resulting from each of the two different sources (Newtonian, non-Newtonian) could be modelled.
Note: As /u/UhOh/ commented, an alternative, possibly superior, equation for the extra, relativistic acceleration is given in the Shahid-Sales & Yeomans paper cited above as Eqtn 3.11, on page 1886.
