Maybe I'm missing something but I don't understand this sentence:
Safety is my primary importance. Would you spend the money on the tire change.
How is a 16" tire safer than a 15" tire - assuming they are both new and they can handle the load rating? I've had a TT for over 20 years with 15" tires and never had a problem - I change them every 5 or 6 years.
Regarding solar - use the batteries that come with the TT; They are new and you don't yet know your power consumption - once you figure out your power consumption then get the right sized solar panel(s);
I have 2x135W panels and they keep the 2 12V batteries topped up during the spring/summer/fall months; Winter I always hook up to shore power to run electrical heaters.