I found the Lizard Skin tape went through to the underlying fabric layer on the bar shoulders the same as the OP, though I got more than 450 miles out of it. Widget tape lasted longer, but not enormously so.
I like silicone tape in use, so I'm now trying T-One Mr Fantastic, as found in my LBS. Drawbacks so far seem to be that it's only in light colours, and it doesn't clean up as well as it did when new. "Have they got any in black?", I asked the LBS man, so he checked the importer website and came back with "discontinued". Daft if you ask me