Can you set up multiple camera views that are side by side and seamless?
Imagine one view that represents the right most 95% of your game window, and the left view is the other 5% of what's off the right side. Then with two views (and a single camera) you can render the same item twice, and create the wraparound effect. Basically when the object is half off the right, it's also being rendered on the left, because the left view represents what's off the right side of the screen.
Here's a pic that might help...
In the above, the green lines are the views, and the black line is the game screen.
Note that it wouldn't be 95/5 really, but based on the size of your object - specifically half the maximum width. If it is 100 pixels wide, your left side narrow view needs to be at least 50 pixels wide.
Edit and Refinement
As pointed out in comments, this does not solve the problem of physics and wrap around worlds. To solve that problem, what you need to do is not think of your world (and its physics) using 2D Cartesian coordinates. Instead, you need to model your world with the coordinate system of a cylinder. A cylinder has no "edge" and as such, there is no concept of "wrapping around the edge" at all. Yes, there is one if you try to go off the top, but if all you are concerned about is horizontal (X) wrap, a cylinder solves your problem.
Here's a picture to show the idea of an object on a cylinder, and how you render it (via something like my 2 view idea above) in a 2D space.
Your only challenge (which I think is valid as a separate question) is, "How do I render a cylindrical world space onto a flat, 2D view port?"
Note 1: If if you want to make things wrap around the top, it's going to get more complex.
Note 2: I am curious how the old asteroids game (which has wrap around on all sides) handles collisions in this (literal) edge case.