Check exactly what the in-game info says for undermining that Power in that system. PowerPlay is complicated, and exactly what you can do to undermine a power depends on that power, your power, and the relationship between the two powers.
If you are undermining a power that's affiliated with the same major faction (both empire, for example), then you have to steal the special cargo. Do not kill the ships, as that will cost you merit points.
You can minimize your reputation hit by following ships into their low wake (instead of interdicting), and being sure to only assault them, not murder them.
Since hatch breaker limpets only work on unshielded targets, you're still going to need to fire at the ship until their shields are down, which will get you a small (~200-400Cr) bounty. (I have no idea if you get an assault charge if you find an unshielded ship and fire a hatch breaker limpet at them) In fact, since you're already firing at them, you can also fire any weapon at their cargo hatch, or severely damage them until they drop some cargo (as part of trying to run away from you). Just be careful not to take their hull or power plant down to 0%.
When you earn merits with your major faction allied Power, I think you also gain some reputation with that major faction. With the assaults (and no murders), it should balance out to a net positive reputation increase with the major faction. (but maybe you should try doing only a few of these robberies, turning in the merits, and seeing what the little arrow next to "Empire" looks like in your status panel).
If you find systems where the minor faction in control isn't allied to the major faction you're with, that should help quite a bit with keeping you from earning a negative reputation with your major faction.
Also, many of the Powers have "oppose Expansion" activities that involve special combat zones. If you go in there, join up with the opposing side (right panel, functions), and kill ships, you earn merits (and oppose the expansion), but I don't think you will lose any reputation with any factions (other than the opposing Power, which you're probably already hostile with). Apparently you can only do this against a Power that doesn't have the same major faction alliance as your own, though.
Also, you could always just stick with undermining powers that aren't connected to your major faction...
Always check what the info says for undermining that Power in that system. You can get to that info from the PowerPlay overview (look at power you want to undermine, find the system and click on it) or from the PowerPlay contact at a station in the system.