Well it's in their hands, but tbh if I had any sway and it were my car, I'd be gunning for the full engine replacement.
They're going to have to pull and re-install the whole engine either way, so any risks from that are common to both solutions (e.g. errors in the installation process). The VIN on the lower engine block is going to change either way, if it's the matching VINs you're worried about value-wise.
I'd rather have a whole new engine in that case than have them slap in a new shortblock + existing heads. You get newer everything for free, and less chance of some unseen fallout affecting the heads' longevity. How well do you trust that their mechanics would notice (or even be *able* to notice) some subtle problem that developed in the heads from running oil-free?
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