Park in Hove opposite the town hall, then bus along to Churchill Sq, tap on tap off for payment. Or you could walk to Churchill Sq in 30 mins or as you are with the kids could do that via the seafront...
I use a bus almost daily going in and out of Brighton and not experienced problems, out of interest what route are you experiencing these problems?
Buses have also started running again along Western Road.
I live in Brighton but gave up driving into the centre about ten/fifteen years ago.