Sorry you're having to go through yet another ailment for Jake, Jakesmomm
It's really tough to watch them grow old.
Have you had some tick panels run? Some of the tick-borne diseases can lower the platelet level and cause abnormal bleeding. The tests might be worth a shot and they shouldn't be that expensive.
A scope and flush isn't too invasive and might be helpful. There's always the possibility that there's something stuck up in the nasal passages that can be dislodged. A scope might also locate a mass
But at least then you'd have a definitive answer.
One thing that we always take into consideration before we run tests--will the results change the treatment being used? With our Priscilla, we treated for discomfort and secondary infection; we ised cool compresses across the snout to help stop the bleeding--but we never did the biopsy because there was really no treatment if it was cancer. The vet could see a vague mass in an inoperable place on a regular x-ray--but since it was inoperable and they told us up front that treatment options were poor, we never spent the money on the CT or an invasive biopsy to confirm that she had cancer. We decided instead to keep her happy and comfortable until her time came. So basically, our decision not to pursue more invasive and traumatic therapies meant that Priscilla's treatments would remain the same regardless of test results--so we skipped the tests.
I know what a difficult time this is for you and I wish you, your husband and Jake all the best.