I'm so sorry about Lucy
My last cat had Lymphoma & Chronic Renal Failure along with possible IBD and near the end a persistant infection. From the time the lymphoma was diagnosed & confirmed until she was pts was only a few months. She was being treated with homeopathic remedies and near the end we added Clavamox antibiotic pills.
Lymphoma & IBD have been linked in some theories, if she had IBD then it's harder to absorb nutrients. Lymphoma is a rapid growing cancer because it spreads through the lymph system which is all throughout the body.
You mention an antibiotic shot - I'm going to assume it was Covenia. There have been suspicions/reports about various reactions pets have had to the shot, as of yet I don't think there is any concrete studies to confirm/deny a link.
It is possible the lymphoma was very aggressive and the shot had little/nothing to do with her death, it's also possible her system was overloaded and being so tiny she couldn't muster enough of a fight.