First, as you will be aware rat poison tends to cause dogs harm by preventing them from clotting their blood. It doesn’t tend to cause the signs of GI upset you are reporting. Instead, the drooling, lip licking, and appetite loss is more suggestive of an unrelated bacterial or viral gastroenteritis, pancreatitis, parasites/protozoa infections, general dietary indiscretions, and ingestion of harmful items (hopefully less likely here).
Now with him being so young, we do need to tread with care. If he is very lethargic and hasn't any energy, you can give him a blood sugar boost just now. To do so, just rub some karo syrup, honey or similar on his gums. That should get some energy into him and hopefully perk him up.
Afterwards, we can try some home supportive care to try to settle his stomach. To start, if he hasn’t just vomited (since otherwise we’d need to rest his stomach for a few hours first), then you can consider treating him with an antacid. Common OTC pet safe options would be Pepcid (More Info/Dose @http://www.petplace.com/article/drug-library/library/over-the-counter/famotidine-pepcid) or Tagamet (More Info/Dose Here @ http://www.petplace.com/article/drug-library/library/over-the-counter/cimetidine-hcl-tagamet). Whichever you choose, we’d give this 20 minutes before offering food to allow absorption. Of course, do double check with your vet if he has a known health issues or is on any medications you haven't mentioned. Though I'd note that if you give this and he cannot keep it down due to nausea that is usually a red flag that we need to bypass his mouth with injectable anti-vomiting medication from his vet.
Though if he can settle with the above, we can try him with a light/easily digestible diet. Examples you can use are cooked white rice with boiled chicken, boiled white fish, cottage cheese, or scrambled eggs (made with water and not milk). There are also OTC vet diets that can be used (ie Hill’s I/D or Royal Canin’s sensitivity) too. The aim of these diets is that they will be better tolerated/absorbed by the compromised gut. Therefore, it should get more nutrients in and result in less GI upset and diarrhea. As long as improvement is being seen, I usually advise continuing this until the signs are settled, and then weaning him slowly back to his normal diet.
Since dehydration is a real risk for pups, we need to keep an eye on his hydration. To check this and ensure he’s not becoming dehydrated, there are a few things you can test. Further to checking for gum moisture, you will want to make sure his eyes are not looking sunken and that he doesn’t have a "skin tent" when you lift the skin. To see how to check these parameters for dehydration, you can find a good video HERE (http://www.ehow.com/video_12232503_dog-dehydrated.html). If you are seeing any signs of dehydration already, then that would be our cue to have him seen before this becomes an additional issue for him (especially as it is often dehydration that makes them feel unwell).
Finally, since his diarrhea is thankfully normal colored and without blood, you can consider trying a pet safe anti-diarrheal. As I am sure you appreciate, these would not be a cure for infectious issues; but it can still be of benefit. It will reduce diarrhea load, allow the body to absorb more water/nutrients, and soothe the upset gut. In regards ***** ***** options, the one we most commonly use is Kaolin/Kaopectate (More Info/Dose @ http://www.petplace.com/article/drug-library/library/over-the-counter/kaolin-and-pectin-kapectolin-k-p)). This is available OTC at most pharmacies. Do make sure to avoid Imodium or Pepto Bismol for him. Otherwise, Propectalin, Canikur, Fast Balance, and Protexin Pro-Fiber (which is available OTC at vets, pet stores, and even Amazon) would be another option. All will slow diarrhea and those last ones have the added bonus of providing support to the delicate good GI bacteria. So, these can be used as a short-term means of soothing his upset GI.
Overall, these signs do not sound related to what you have recently gone through with your other pup. Instead, this all suggests an upper GI issue and therefore in his case, we’d want to start supportive care to settle his stomach. If he cannot keep that or water down, appears dehydrated already, or doesn’t respond to the above within a few hours (since he is a pup and is lethargic with this) then we'd want to get his vet involved. They can assess his hydration, rule out fever, make sure there is nothing in his stomach that shouldn't be there or any sinister viruses present. Depending on their findings, his vet can treat him with injectable anti-vomiting medication, fluids, +/- antibiotics to settle his stomach, and get him back feeling like himself.
Please take care,
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