Here, in all humility is my suggestion:
Rats Action 2$
+1 Card per Rats in play, +1 Action
If there are 2 Rats in play, immediately put them into your discard pile and then reveal your hand, if you have no action cards in your hand, gain a Rats.
My thoughts are this:
1. 1$ cards create problems, I think creating a balanced one is really tough, keep it at 2$, it becomes easier to measure and contrast with other cards.
2. The "fast depletion" aspect of Rats is both a feature and a problem. It's a feature because its one of the key reasons someone would want to invest in the card, also its thematic that you wind up littering your deck with them. I like thematic cards, so trying to make the depletion aspect of the card work, is something I think is worth solving. But the depletion feature is a problem, because you can't make that depletion too fast, otherwise the first person to chain two, will likely deplete the pile very quickly.
This card provides solutions to both problems. It sticks the card at 2$. Now you can decide if you want a Pawn or a Rats. This is useful, it helps you judge the card easier, (both in-game and at the design phase).
In fact, the first Rat is a basic Pawn (the most commonly selected utility of Pawn anyway) and the second is a Lab. That's it, that's all you get, then you discard the Rats and start the process over. That's not stunning for two 2$ purchases, but it isn't horrible. But what you really get is a cheap, effortless gain of a Rats (assuming your hand has no more Rats in it). This is the appeal you want. You want Rats to be a bit cumbersome, but they get you to a 3-pile condition, which at times is a sublime way to win.
This isn't the most elegant solution, but I think it might behoove you to think more in this direction.
The solution here makes the free Rats gain, on the condition you have none left, which really governs it to just one Rats gain per turn (at most). This helps govern the "lottery" effect of chaining early and often to deplete the entire pile. Also it pretty makes having an "odd number" of Rats slightly less appealing than an even number. You get to the point where on the second Rats, you kind of hope you don't draw the third, otherwise you may need to spend a buy getting a Rats. There's some playfulness there I think you might enjoy.