If a dogs had a reaction to a vaccine the chances are very high they`ll have another one to any vaccine they receive afterwards. Vaccines should only be given to healthy animals & as your pup is unwell I don`t see how another vax can be given, your vet & the manu`s know this

The usual first puppy vax are distemper & parvo not lepto, although some also contain lepto as well, maybe check & see what they`ve had. The vax are`nt *smaller* just either 1, 2 3 or more different viruses.
It`s a *suspected* adverse reaction, nothing has to be proved to report one & you can even do it yourself if you want to & your vet won`t.
Sorry about your pup tho, hope he gets better soon, try & get a firm diagnosis, if it was me I`d take him to a homeopathic vet.
Good luck :)