When it comes to foreign vape laws, I'm a bit afraid of giving wrong advice and getting someone in trouble, so I urge you to do as much of your own research as you have time for.
This page says there is no specific vape law in Belize
138 people die every year due to tobacco smoking in Belize. tobacco smoking-related health problems account for 8.37% of deaths There is currently no specific law regarding e-cigarettes – e-cigarettes...
gsthr.org
So does this one:
Are you planning to take your vape pen to Belize? Find out the latest rules and regulations for flying with a vape pen, so you can enjoy your trip without any unexpected surprises.
knowyourvape.com
Therefore, little chance of having your gear confiscated when you go through Belize Customs.
You're not supposed to put batteries in your checked luggage, but the times I've flown with vape gear I've had no trouble having two batts in my carry-on, and as you mention, the 100 ml. liquid in the silly quart size baggie with your other liquids such as shampoo, lotion, etc. You can put more eliquid in your checked luggage, where nobody will care about it.
www.tsa.gov
I hope you have a wonderful time and find a good place to buy.