The short answer is no, that's not nearly long enough for late stage issues. Before suggesting it to your doc, take a look at the treatment guidelines from ILADS:
goo.gl/QyiAhd And this is what Dr. Joseph Burrascano said about
ceftriaxone use (bottom of page 16 here:
www.ilads.org/lyme/B_guidelines_12_17_08.pdf):
"Treatment with ceftriaxone is dosed at 4 grams daily- given either as 2 grams IV twice daily, or 4 grams slowly
once a day, four days in a row each week, usually for 14
or more weeks. Such a regimen is not only more
effective in the Chronic Lyme patient, but regular interruptions in treatment lessen the potential complications of
intensive antibiotic therapy with ceftriaxone, such as biliary sludging and colitis."
I would hate to see you go through the possible issues with IV use and the cost of it all, only to find that you have relapsed shortly after stopping treatment. It's best to make sure you get the right amount of treatment from the beginning.