Congrats on your new coach and welcome to Sunline Owners Club!
The T2499 is a popular model. We used to have one before the T310SR came. Ours was a 2004 model.
Your spare, that 14" tire is not correct. It should of been a match to the 4 main tires. It got mixed up some how on you.
I looked up the ST 205/75R14, you did not mention the full tire number so I assumed you are on ST's and radial tires. Please check.
There is another piece of very important info, what load rating that 14" tire is as they make several load ratings in that size. Look on the side wall for what load capacity the tire can hold at full cold pressure and what is the pressure?
The ST205/75/R15 that came from Sunline was a C load range and could hold 1,820# at 50 psi which is the proper size for the 7,000# GVWR of your 2003 T2499.
The ST205/75/R14 comes in 4 ply tire at 1,480# at 35 psi (not good at all)
The ST205/75/R14 comes in 6 ply tire at 1,760# at 50 psi (still a concern)
The OD of the 2 tires (14 and 15") are different as well.
Another thing to check is the tire DOT date codes. See here on how to decode the month and year the tire was made. You could be on 9 year old tires.
Tire Tech Information - Determining the Age of a Tire
And look in the threads of the tires and the side wall for cracks. The cracks can be small to big but they are still cracks. It is common when trailer tires reach their 5 year old point that cracks can come. Some get lucky and they take a little longer and some get unlucky and they happen a lot sooner..
You can get a wide range of answers when trailer tires are suppose to be changed. I myself go by the 5 year time period and for sure if cracks are forming. Unless you really tow a lot, most times tires age out and never wear the thread down.
A tire blow out can cause a lot of damage on a TT and hopefully not upset the tow rig. Again some get lucky and only have some black marks to scrub to death to get off. Others are not so lucky and end up replacing the fender well, fixing the water vapor membrane and needing a new fender skirt.
I'm on the conservative side of towing, all tires including the spare get changed when the age/cracks come. With my luck I will be out in the boonies if a flat comes and I want to count on the spare to be as good as the rest.
Good luck, hope this helps and happy camping in your new coach.