first, I think it's probably a bit creepy to have a picture of Sue Johanson on the box. I don't have this one, but I'd be interested to try it out given its design.

second, I think you should probably let the toy warm up in a sink full of hot water before you attempt to clean or use the toy. TPR will get more pliable and less likely to form micro-fissures if it's warmed up. Heavy use without warming has the potential to cause these sort of issues. The design of the toy is lips with creases in it. I have never used it, but I'm assuming that the corners of the lips is where it splits. It may just be a design flaw that is unavoidable.

Some of my toys are too thick or fragile to turn inside out so I just pump the soap into the tube and use my fingers, then I hold it up to the faucet and hose it out. This is more effective than you think though you have to air dry it before storing.