I've been using head shots on rats for several years, using a Pardini K12 so precision placement isn't a problem. A piston rifle like you're using can indeed be very challenging to shoot accurately - the 'artillery hold' being the rule, let the rifle move a little as you fire instead of trying to restrain it. Lots of practice in daylight at different ranges will tell you what's working, what's not, and you can establish a range card for these closer ranges. With a PCP the gun doesn't move at all, just point and click. I have a little NV200 infrared monocular set up behind a 4x scope (the infrared flashlight element extracted and raised above the scope turret and put into a tube with a focusing lens up front) and can take them anywhere between 2 yards and 15 yards with confidence, beyond that I would worry the low power was insufficient.
I'm using JSB round nose pellets in .177 weighing 8.43gr, shooting at a little under 500fps. I had the pistol for competitive shooting and when I stopped doing that owing to a shoulder injury this was the next role for the thing. I find that for head shots a sub-500fps airgun is plenty, body shots I'd probably want to go for a PAL-rated airgun, something around 700fps in .22", and still trying to get the heart if possible. But on a rat that's likely harder than a brain shot, so I don't really see the point.
Close range holdover is the most challenging part, with head shots requiring placement within less than a dime sized target to ensure a quick drop. I usually go for side of the head, slightly in front of the ear hole, but if the rat is facing me (they often are, as they can see the dull red glow of the IR LED) it's usually with their nose down, so I can get a decent forehead shot. Haven't taken a body shot since my first few rats as though such hits worked, it wasn't an instant death. I really don't want them running off where I can't find them, having rotting rats laying around.
For my setup, with fairly high-mounted scope, I have to shoot about 2" high for point-blank (2 yards or so is as close as I've ever got one), about 1" high at 7 yards, 1/2" at 10 yards, zero at 15 yards. Every spacing between scope and bore will make for a different trajectory of course. I tried a Torrey Pines (now changed to Sector Optics) T20 thermal unit for a while, but the screen resolution just doesn't offer precise enough reticle adjustments to stay on a rat's head beyond a few metres. Too bad, as a rat has no idea you're looking at it with a thermal scope. So back to the infrared, which I have to turn on and shoot within a few seconds or they spook and run off.