There's probably plenty of ways to get there from adding additional physics. Specifically ruling out relativity is probably extremely doable. That said, if you want to ignore other observed phenomena and focus on just this orbit without adding anything else... then possibly?
Since this isn't hard-science I'm going to throw a thought experiment at you.
If you imagined a redistribution of mass in or near the Sun to account for this orbit then you have the same question about this mass distribution:
"How did you get it with Newtonian physics acting on itself?"
Which you could maybe answer by saying there was some coupling on it of a similar nature. Which maybe Mercury or the new mass itself provides. If you're providing a closed loop of effects at some point then you're positing a "stable" system in a now extremely sensitive and chaotic environment; since it's an orbit not predicated on a single giant mass with negligible outside interference, but a "chain" of masses in concert. Granted it's less chaotic without things like gravity waves, etc. but I would... assume that orbits would decay faster than they already do for such a precisely orchestrated setup. Unless there was some self-reinforcing factor for this distribution of energy and mass.... which, under Newtonian physics, I don't recall there being any mechanism to achieve this in space. Stability under precise conditions maybe, self-reinforcing stability I don't recall though. Maybe balancing pressures (light, etc.)?
Conclusion, don't necessarily know enough off-hand and didn't do the research. But it seems like possibly with a decent dose of "I am suspending my disbelief for you". (Especially since any world building where mentioning this as a factor in-universe is going to have them thinking about ramifications of non-relativity anyway... ie readers are smart too)