If you add all the liquids and a big solid lump of peanut butter, you'll have a tough time getting it all smooth, as stirring the (thin) liquid parts won't affect the (solid) peanut butter lumps, and the lumps (once they're small enough) will just swim around your spoon and not break down further.
You need to gradually dilute the peanut butter with the other hot liquids, stirring each time until the mixture is smooth. That way at each stage the liquids won't be vastly thinner than the peanut butter, so stirring will properly mix the two. Once the peanut butter mixture is thinned to the consistency of, say, mayonnaise, you'll be able to add the rest of the liquids.
It's the same principle when making gravy from flour, fat and broth. If you add all the broth at once to the roux, you'll end up with lumpy gravy. But, if you add a bit of broth at a time, stirring until smooth each time, you'll never have lumps.
To fix your problem in hindsight, try running it all through a coarse sieve, or use a potato masher on it.
The easiest solution that comes to mind would be to figure out how much you need for a sandwich, remove that from the jar, set it on the counter to come to room temp (or put it in the microwave for 10-15 seconds) and then spread it when it's warmed up.
"Refrigerate after opening" doesn't mean that it must be in the fridge every second, though... so even if you left the entire jar on the counter for 30 minutes to warm up, it shouldn't pose much of a problem.
Best Answer
One solution is to make your own peanut butter.
If you have peanuts (preferably roasted; if you have raw peanuts, toss them in the oven on a baking sheet in a single layer at around 350 F for about 10 minutes, maybe with a tbsp of peanut/vegetable/canola oil), throw them in a food processor or blender and let it rip. Scrape the sides of the bowl periodically. You'll (eventually) end up with peanut butter.
If you want chunky, run the food processor until the peanuts are chopped, reserve a portion of them and then run the food processor until the rest is smooth. then, put the chopped peanuts in and pulse to combine (or just stir them in).
Season with salt and sweetener (e.g. honey) to taste.
For more details, just find a recipe like this one from The Kitchn or this one from Alton Brown.
Note that this is going to be relatively expensive versus just buying a jar of peanut butter.
The usual substitutes for peanut butter are other nut butters like almond butter. See this question for some common substitutions. Or, make a different type of cookie.