The "more accurate method" will be to either numerically integrate orbital motion to make your own approximate ephemeris (see this and this answer) or as @Vince49 points out, download and interpolate existing ephemerides, in this case for a very large number of minor bodies! Neither is a pretty or elegant solution but because gravity is long ranged and "everything pulls on everything", Keplerian orbits won't do here.
Realistic solar system orbits are only approximately Keplerian.
I'll mention that integrating yourself is fun and educational if you really enjoy that kind of challenge, but doing it right is much more complicated than shown in those linked answers. I was able to match the JPL ephemerides to 1 km over 1 year for a dozen large bodies, but without correctly addressing the tidal effects between the Earth and Moon, the Earth's position will continue to degrade each year. You would need to do a more sophisticated calculation than the approximation shown there.
For smaller bodies, non-gravitational forces such as radiation from the sun and by the body itself, as well as outgassing when near the sun are also important. Those can be approximated as I've described in this question and in this answer.
So I think it is likely that sooner or later you will probably decide to use existing ephemerides. If you like to program in C, there should be documentation out there to use things like the Spice toolkit to interpolate JPL kernels. Once you get good at it you may decide to construct less accurate much smaller tables to interpolate, considering you may need to do thousands or tens of thousands of minor bodies.
@RoryAlsop's excellent answer includes several helpful links, including a Python library jplephem that interpolates Spice kernels. This might be particularly helpful to you.
The Python package Skyfield is already set up to automatically download and interpolate the JPL Developmental Ephemerides, but these are only for the major solar system bodies. Since you are already using Python, I'd guess you would really enjoy reading through that package. It's a joy to use.
Also, the Python package PyEphem performs a wider range of orbit functionality for you. I'm less familliar with it so you'll have to investigate yourself. It's been around quite a long time, and I have found out is based on XEphem and VSOP87 (and see also).
The Python packages SkyField and PyEphem are both supported by the same person.