I have a whole lot of experience with rats, big and small. I have had some as you describe...they just wouldnt die. I shot one where the pellet scalped him, took an eye and a little skull with it. He ran away!!! I coulnt find him until the next morning...floating in one of my ponds. These things are tough. Shot placement is everything. I've seen em go down without a twitch, and further shot analysis showed brain shots or neck/spine severance. The dang things are so spastic, that what spot you aimed at when you pulled the trigger aint quite where the pellet lands.
I had the most luck with a "bait station", a red light for illuminating the station(2D LED flashlight with a red filter), a motion detector(driveway patrol on ebay), and close range shots. If you dedicate a portion of your deck to rat killin you could beef that portion up, reinforce it so to speak to take them .25 shots...line awooden box with duct seal...and put the bait in there. You locate the bait station in such a way that you have clear, close shots...without the rat knowing you are there. For example, I set up a bait station under the kitchen sink window. Got the screen off the window, sighted in the gun for the 6~8 foot shots that I would be taking. Left the window open a crack, enough to let the barrel out and get the shot out. Motion detector went off, I very carefully went to the window, and squeezed off a shot at the rat. None escaped!!!
Now, I realize that you may not be able to get that close...but there's gotta be a way to set up a bait station somewhere and turn your situation into a sport. Why not have some fun? You don't have to use a Falcon...but if you did, it would be awesome, and shot placement anywhere in the center mass of the rat would either take em out right there or very quickly thereafter. The good thing about a bait station is that they tend to be way less spastic while eating!!! The rats got smart and started dashing in to take a little food and then dashed back out, so I put the food under some screen so they had to try and get into the food. While they were pausing every few seconds to figure out how to get to the food, a pellet went into their thinking area, if I was lucky. The trick is to bring the rats to you...not you go to the rats, that way you can control everything.
I think you should get a Falcon .25 anyway...just so you have it.