DPC every time - even if the base has a dpc, because the top of the slab unless completely covered can still wick moisture and rot the underside of the wood.
There's really no reason not to, and trust me when I tell you jacking up a side of a wooden structure enough so you can cut out a rotted runner ("Base Plate" as its known) and replace it is no easy task; been there, done that.
I would also go with company B - using BP's under the "joists" which I am assuming are referring to the vertical supports. Using a BP has 2 benefits - 1 is you can bolt down the frame to the slab, the other is should a BP somehow rot or be otherwise compromised, it can be replaced - a real faff but not impossible. If no runner is used and the bottom of a vertical support gets compromised... well that's some fairly complicated surgery for an average diyer to fix.
BP's also help stop the bottom of the supports from splaying over time as the wood moves, and it will.
TBH I can't imagine why any wood framing company would even suggest NOT using BP's - in the 20 odd years I've been watching american TV of that genre, every single wooden framed building I've ever seen built, and that's many hundreds if not thousands, I can't recall ever seeing them deliberately not using a BP; it brings so many benefits and the extra cost is minimal when considered to the overall expenditure.
If you do go with BPs (and I strongly advise you do) and they are being bolted to the concrete slab - drill the hole through the wood oversize and essentially fill the hole with silicone before inserting the bolt and washer into the slab - this should greatly reduce or prevent water wicking up the bolt from the slab and into the wood where the bolt compromises the dpc under the wood. The fact there is minmal slab exposure still won't stop the slab getting wet even if you plan to clad the sides of the slab as well, because it'll still absorb ambient moisture.
Drilling the holes oversize will also allow the wood to move over the concrete, and the concrete slab to move according to the seasons as well.