Tao Tao 49cc ATM50-A1, tail light issue advice

[SpiritOf76]

Hi - my name is Steve. I'm a new forum member enjoying a used 2012 TaoTao ATM50-A1 that I got a good deal on. It does have some things that need help, though.

Does anyone have an idea what would cause the tail light to light up when the scooter ignition is in the on position, but then go out when either brake is grabbed? TIA for any ideas.

Also looking for a good battery to replace my current one, which can't provide 12V anymore; it tops out at 9V after a few long trickle charge attempts. I did see a couple of links in the forums search, which should be of help. I guess I'm looking for a 12V 7Ah or 9Ah that has a good CCA rating and I'd like to keep the cost under $50. I plan to take the battery in the house for the winter, but put it back on the bike on the occasional nice winter day to start the bike and run it a bit.

TIA for any thoughts.

- Steve