Objects flying into the car is definitely a risk, but I remember reading here at least once that convertibles have lower rollover risk & their body structure (aside from the soft or open top) is usually quite safe because they're basically reinforced to compensate for the open top ... so side impacts are probably less of a concern.
As long as kiddo is safely & correctly restrained in a 5-point harness carseat I would be OK with it for the occasional fun trip (I'd prefer the top closed, but if it's kiddo & dad's rare chance for time together I'd try not to worry too much). Hopefully a technician here will say if I'm wrong
Also, what is kiddo's weight & height? Is s/he still Rear-Facing? Rear-Facing is safest, especially if kiddo is in the front seat & reverting kids back often isn't as big a hassle as parents imagine ... there are also carseats that offer extended harnessing instead of graduating kiddo into a booster too soon; we can help you choose the safest possible carseat for this situation when we know kiddo's weight & height