I don't know...I have no experence of "split heat"
My dog was "entire" for 12-13 years. She was spayed last year owing to acute Pyometra.
In all her heat cycles, she never had one heat on top of the other, so in my opinion, your dog's bleeding sounds abnormal....but then again I can't be sure as I have never encountered "split heat"
It could possibly be hormonal disturbance owing to her youth? Similar to the irreguar periods which often happen just after a Human's periods start, (and which can also appen at menopause) as the hormones are swinging wildly at that age, until they stabilize.
However -that is just a guess on my part. I really am unsure.
Usually Pyometra presents very suddenly (though not always!) with loss of appetite, lethargy, sometimes vomiting, and sometimes vaginal discharge which can be greenish, can be foul smelling -or even sometimes, as was the case with my dog -mucusy and white, with no bad smell at all. The worst and most dangerous form of Pyometra is what is called "closed" Pyometra, where there is no vaginal discharge at all. The reason this can be extremely dangerous is it can often be mistaken for upset stomach, or a host of other complaints, and valuable time can be lost with mis-diagnoses.
Usually a dog with Pyometra will act "sick".
I say usually....there could be exceptions. It is possible.
I totally understand your unwillingness to subject her to a whole lot of tests, especially as she seems very well. A reputable and decent vet certainly wouldn't deliberately rip you off, but if one test didn't give definitive results, another would have to be run to eliminate the possibility of disease -and so on.
There would be no harm in taking her for a urinalysis and blood test. Both could show whether there was any systemic infection present. Altogether, including consultation, that might cost you around the $200 mark (?) (That's a bit off the top of my head, because here in the UK my vet would probably charge between £120-£150 for those tests)
It might be worth doing just to set your mind at rest.
Please excuse my typing errors. This keyboard is terrible.
In a split heat, the first heat is a false heat and only lasts a week or so, then the true heat comes along a couple of weeks later and is normal. In the false heat there is no ovulation, no flagging, the female will not let the male breed her. Usually split heats occur when there are multiple unspayed females in the same house. This phenomenono occurs in wolf packs so that all the females are in heat at the same time. Only the alpha female will be bred and have a litter but with all females being in heat at once. This ensures that if the alpha female dies, there will be other females who can be stimulated to lactate to feed the alpha females litter. It COULD, however, be because your dog is so young.
Just to be on the safe side I would have her checked out to make sure she's not having discharge because of a pyometra. It will be worth it to make sure she's OK.