Its a halfords own make 3 cycle rear mounted i have the now, but I only ever have just my bike on it, the straps u use are just the nylon straps that came with it, would elastic hook straps be any better
Spinney's suggestion is a good one, although I'd suggest re-usable cable ties to hold it in place as it's a pain if it blows off.
Plan B would be some clear plastic bodywork protector that you can get, usually for motorbikes, like this stuff. It's like heavy duty clingfilm and it's almost invisible to the eye, I have a few bits strategically placed to stop cables rubbing.
Nylon straps on their own should cause an issue. Are the wheels dirty, because dried mud will scratch if you strap over it.
I'd avoid the elastic cord for holding stuff on. It's ok for stopping bikes from flapping and wheels from spinning but, if you are to get it tight enough for security, there'll be enough tension in it to cause significant pain if it slips. And I'd never use the hooked straps anyway, they can scratch any bodywork they touch and they'll remove anything that they catch on if they slip. Either use Rok Straps or something like these, which are both safer and more secure, IMO.
I am not disagreeing with Sheffield Tiger about using ratchet straps, but do be aware that they can exert a huge force so be careful that they don't buckle any part of your bike or your carrier. Used wisely, they are good but I have seen people use them unwisely with potentially expensive results.....