Unless the battery is absolutely knackered, it should retain its charge for weeks. Unless, there is an underlying problem elswhere that is causing it to discharge. In which case, any another battery fitted is just gonna suffer the same fate.
Try using an ammeter (10A scale on a digital multimeter) connected between a disconnected battery earth lead and chassis. If you are pulling current with the scoot at rest, then further inveatigation is needed. (A reading of a few microamps is normal due to leakage current in the Zener diode inside your regulator.)
If you haven't got a multimeter to hand, I would suggest charging the battery up, and then either disconnect the earth lead or remove the fuse and leave it like that for a few days before connecting it back up and checking to see if it has held its charge.
The shunt type regulators on our Lammies are rather primitive and have a propensity to overcharge batteries. Not so much an issue if you only tootle around town, but causes the battery to overheat, dry out and distort on high mileage touring steeds. This severely shortens their working life if they are a not regularly topped up with distilled water, which you can't do with a sealed lead acid "Gell cell" battery. However, running your scoot with the lights on all the time, reduces this overcharging affect on your battery to negligible levels!
Sealed batteries can be made to work very well offering life spans exceeding 10+years, but ideally, they need a better, more controlled, "series type" regulator.
A GB5L-B Gel-cell battery will fit in the standard battery tray but only has about 60% the A/H capacity of a comparable wet cell battery, which is sufficient on most applications.