I'm trying to learn the connection between special relativity and magnetism. I know that if I place

The electrons themselves are length-contracted (not that such an effect is measurable). The distance between the electrons is not contracted. The density is constant in this frame.
This makes sense. Regardless of the speed of the electrons, the circuit path is the same length in this frame and the total charge (number of electrons) is fixed in this frame, so the density of charge remains fixed.

Consider a wire with no current flowing through it, and equal densities of negative charges (electrons) and positive charges (protons). Give the electrons some velocity (such that, in their rest frame, they have constant separation): then you will get a current due to the moving charge, and you will get a negative charge density, due to length contraction. This reasoning is correct. The wire now carries a current and has a negative charge.
Consider a wire with no current flowing through it, and a lower density of electrons than protons. This wire has an overall positive charge density. If you give the electrons a correctly chosen velocity (again, such that, they see no change in separation), then you will get a current due to the moving charge, and the negative charge density will rise (due to length contraction) to cancel out the positive charge. This reasoning is also correct, and this time you have a wire with a current but no overall charge.
You see that the current through a wire doesn't determine the charge on the wire: you also need to know the charge on the wire when it has no current. By adding or removing electrons from the wire, you can get any combination of current and charge (in principle—real matter will probably disintegrate at some point). In your scenario, the wire is constructed as in my second example, so that it carries a current and has no charge in the frame of the test charge. The test charge thus feels no force since there is no electric field.
Note that real wires are more complicated than this. Consider a loop of wire with no current and no charge. Apply some force (visiting bar magnet?) to get all the electrons circulating through it, like the first construction. In this scenario, the electrons can't see a constant separation in their rest frames, since that would imply contraction of the negative charge in the rest frame of the wire and the separation of charge into positive and negative zones. The density of electrons in the wire's rest frame must remain constant, so the electrons in their own frames see the other electrons pulled away from them. The end result, once the force is removed and we have circulating electrons at equilibrium, is that each small segment of the wire "microscopically" looks like it was made by the second construction, as the overall charge must remain zero. If this sounds weird, it probably should. Consider the Ehrenfest paradox and Bell's spaceship paradox to further understand how this works.