Contact resistance between pogo pin and target will depend on a number of factors including head shape, spring force/deflection, target contact material, platings, surface cleanliness/oxidation. Most of those are out of the control of the pogo pin manufacturer, which is why you don't see much in the way of resistance specs for them. This is also why you want different head styles: a single needle-point pin will have higher point pressure and will therefore better punch through surface contamination/oxides on the surface, so are a better choice for boards where the target surface is questionable. But the point tends to leave a small conical divot, which may be objectionable, especially if you're making contact with the same pad repeatedly. At that point, you might be better off with a spherical tip pin and a gold-plated metal target pad, which will be less tolerant of surface contamination but won't damage the pad nearly as much. So it's a bit of a tricky question.
I don't think you're going to find a pogo pin good for 5A, mostly they top out at 1-2A each. You can of course cluster the pins for higher currents, but that means more force (all of those springs add up). You might find SMT spring terminal blocks for batteries with more suitable per-contact ratings.
Do you really need to handle all of the current via the one contact? Or could you not use the contact/switch to control a relay or transistor to handle the load current?