I have the same kinds of rims, with tube tires. I did a lot of research on tires before buying a bus, and finally decided that as long as the tires themselves were in good shape, it really didn't matter. Split rims are different to work with, and require the proper equipment (safety cage); however, many tire shops that deal in tires to fit buses and trucks have such equipment -- it's not like you're going to be doing your own tire mounting -- so that's not really an issue. If you have a flat on the road, you're most likely going to call a service truck to fix it, so handling the wheel is probably not an issue. The "danger" from split rims comes primarly in the process of mounting and tires on the rims. Once the rim is on the bus, it's no more likely to than any other rim to be dangerous (practically speaking).
Tube tires are slightly less reliable than radials in some regards, but, again, for practical purposes (such as tailgaiting or camping), they are probably not noticeably less reliable than a radial.
I bet you'd spend a minimum of $500 or $600 to replace all your tires and rims (if you could find a used set in good condition), and you could easily spend much more.
On a cost-benefit basis, I'd suggest that your money would be better spent on other things. Of course, if you have plenty of money to throw at your bus, and you want some nice new rims and tubless tires, there's no reason not to get them. If you do replace them, you might consider getting the single, big, wide 14" tires and rims in the rear instead of buying another set of dually rims. The single, wider tires are supposed to give better fuel economy and a better ride. Plus, they get you around some restrictions on vehicles with 6 or more wheels!
There are plenty of split rimmed, tubless tired trucks and busses running around without problems. Lots of the schools still use them, and if the things were as dangerous as you have been told, the schools would be replacing them all with newer equipment.
If I had a jillion dollars to spend on the bus, I'd get a 6 speed manual transmission, a lower rear end ratio, and a set of wide rims for the rear.
Among other things!
This is just my opinion. Other folks might see it differently.