The LIKE200i is quite famous for its super quick starts in all temps - including VERY cold! - even with the stock 8Ah battery
(would be nice to have some more info on your scoot when problems are posted: year, miles, recent repairs, wrecks, ? etc.)
So, I am very suspect of your battery. EFI likes a nice healthy battery. Judging by a decade of reports here, Starters on the LIKE200i are fine. Not at all a weak point in the starting system....unless very sorely abused (like any starter)
I'd certainly try a "known" good battery before purchasing any more parts for your LIKE200i - seller saying battery is good is not much of a guarantee..
When you do replace your battery - think about stepping up to something more than the stock 8 or 9Ah stock battery.
(10, 12, or 14AH)
I have great starts with my LIKE200i - always - and have for 7 years. I did ride my LIKE200i on the highway once, for 15 mins - about 5 years ago.
It is not necessary on your scooter. Ride it normally, put it on a battery tender if it's going to sit for more than a few days.
More $ spent on a better and stronger battery won't make your scooter run any better - but it sure as H will give you a smile when you hit the start button - miles from home on a cold day!
If your otherwise healthy LIKE won't start & run properly with a new, better, (and yes - more $$ - but less than a tow!) battery - then you might want to think about getting your charging system checked.
Stig