I understood you had the factory 240 just wanted to clear it up if anybody else was looking up specs and didn't understand the difference.
Do you know if the pump itself (not motor) is a Jacuzzi also ?? Any ID tag on the pump ?? Is there anything at all on the motor for manufacturer or ... anything ??
Soooooo. having a day to think about it in the back of my head what is happening is an added 1.3 amps is tripping the 15 amp breaker. I'd hate to see you buy a 20 amp GFI and in a couple weeks or months be tripping the 20 amp !! I have to assume whoever put the motor and circuit breaker in would have left more than a one amp buffer between motor need and breaker size. I think there is something starting to go out on the motor. That is just MHO from reading and rereading your post.
A new capacitor might lower that initial draw if the present one is starting to break down. If the motor is really old that would be a pretty good guess, just not a guarantee. How long have you been there ?? The down side is that it would be a matter of buying one to find out. I don't know where you could get one tested. If you know or look up (youtube) how to do one the part itself isn't that expensive. Needs to be the correct P/N for the motor but the Hayward ones for my 1.5hp motor are around $20 on ebay.
Then there is the possibility of a bearing in the motor being affected by age or moisture. That's why I mentioned looking under the pump or motor to see if you can see the remains (dried calcium trail) of a small leak. Maybe even some water dripping ?? You would need to take the motor loose and feel the rotation of the shaft for any binding or grinding. If so, rebuild or replace.
Then there is the breaker. They can get weak over time but it's fairly rare to have one go.
As for the 12 amp reading, yes if you get 12 amps on one leg you have to get 12 on the other. The problem is the meter you're using. I'm guessing you have the average under a hundred buck meter and it might be averaging or taking an RMS reading. To do that kind of diag on a fuse or circuit breaker blowing you almost need an oscilloscope or something like a Fluke brand meter (few hundred $$) that records a spike. Problem with the average meter is that it's still going up and trying to catch up when the circuit has already tripped the breaker and the voltage and amperage in the circuit are collapsing.
Before your meter can read the maximum the circuit is already dead.
I doubt it's actually tripping at 12 but if by chance you do have an expensive meter that's capable of reading spikes and you know it's reading the spike then you do have a bad breaker.