I've been driving around Belize, including remote corners of the country, for 18 years, and the worst thing that's ever happened to me is that on one trip I parked my rental car on a deserted rural road overnight, and somebody tried to break in. They failed, but the lock was broken, and it cost me US$50 to get it fixed.
Personally, I think a car rental is a great way to see the country. I don't think there's any real concern about safety when driving -- just use common sense.
As to the Kendal Bridge near Maya Centre, the temporary bridge was out again a few days ago due to heavy rains. Not sure whether it's open again yet or not. It's been washed out several times since the original bridge was washed away. I would think it will be another year or longer before the bridge is permanently replaced. You should check in advance before driving to Placencia to see what the status is.
It usually takes me about 3 1/2 hours to drive from Belize City to Placencia, but it depends on how you drive, how often you stop, and road conditions.