First off, your tyre supplier is talking nonsense as the 350Zs have never been supplied with any of the Yokos, only ever Bridgestone as I suspect you already know. Never used Advansports myself, I did look into them briefly as an alternative to the MPS2 however the extra cost put me off. From your post I suspect you want tyres at the cheaper end of the market, so if that's the case I'd beg, borrow or steal the money to get the VUS. I'm aware of the supply problems, but unless you're desperate that's where I'd spend my money.
That said, if I've misjudged it and you're looking at tyres in the Advansport price-range, then you'll not find better than the MPS2. The RE050As are very close as well, and I suspect you're getting dodgy wet handling characteristics because you've got mixed tyres front and back: Swapping to RE050As on the front will cure the understeer you're getting in the wet.