For the stated budget, I'd recommend an MM cartridge (or 2 or 3). Put as much of your $3K into the phono stage as you possibly can, you'll never regret it.
I've heard (and own) $200 MMs that compete easily with any sub-$1K MC, and most sub-$2K MCs. The extra $500-1500 you'd have to spend for a decent MC won't get you any better sound, because that money would come right out of your phono stage budget. At this price point, an extra $500-1500 invested in a better phono stage will provide much more sonic performance than the same money invested in a cartridge.
I've heard (and own) $200 MMs that compete easily with any sub-$1K MC, and most sub-$2K MCs. The extra $500-1500 you'd have to spend for a decent MC won't get you any better sound, because that money would come right out of your phono stage budget. At this price point, an extra $500-1500 invested in a better phono stage will provide much more sonic performance than the same money invested in a cartridge.