Lethargy and bloody diarrhea can be caused by any one of a number of things. In a young puppy, Parvovirus would be the first suspect. Coccidiosis and other intestinal parasites and protozoa can also debilitate a puppy, especially a small breed puppy, pretty rapidly. Brown recluse bite would probably not make the top ten list of possible diagnoses with those symptoms.
Parvo or intestinal parasites are the first thing that came to MY mind, also, as someone who bred dogs for decades. Pooping blood is a primary symptom of Parvovirus, as well as worms, and Parvo is almost always fatal in puppies, regardless of breed, but in a tiny dog it will be especially lethal. Nothing I've read described at all sounds like a spider bite of ANY kind, and honestly, if that was MY vet who'd suggested that a Brown Recluse killed this puppy, I'd be looking around for another vet ASAP! A "little red mark" could have come from anything-a flea, a tick, an ant, or even something not at all related to any invertebrate, like a puncture from a thorn or something or a reaction to something in the environment. To connect that with the puppy hemorrhaging internally is really quite a stretch, and Parvo, Coronavirus or worms(especially whipworms) would all be very likely candidates. If the puppy was young, its immune system is not going to be fully developed anyway, in any puppy, and often people vaccinate puppies too early, which means that lingering maternal antibodies from their mother destroys the vaccine, and when those maternal antibodies wear off, the puppy is left with no protection against a host of highly contagious diseases. If it WAS Parvo, it would be very important for the OP to know that, since this virus is shed in feces and vomit and remains in an outdoor environment for a long, long time, putting any other puppies brought into that same environment at risk of the disease.
Parvo is like the Influenza virus; there are multiple strains, and the virus is constantly mutating, so that vaccine manufacturers have a struggle to second-guess which strain will hit what area and to try and stay one step ahead of the virus's ability to mutate. This makes effective vaccination difficult at best. A vet might buy a stock of vaccines only to have a mutated strain not affected by that particular batch of vaccines break out, so that even vaccinated dogs get sick, and of course, there's that issue with maternal antibodies in puppies that I mentioned earlier. There are some totally new and as of yet, undiagnosed viruses making their rounds through certain parts of the US, especially in New England and the Mid Atlantic states, which have no vaccine preventative and which are proving especially deadly. These originated in "rescue" dogs imported from impoverished countries, to meet the demand for such dogs in those areas. Pennsylvania is a really hart-hit state for one of these unknown viruses, and Ohio is right next door.