Here's a picture with one of my chicks, showing the hardware cloth we used (I still have to run some down further on the bottom, just in this area), just to give an idea that we're talking about the same material:
I'm not sure about gauge. I think it MIGHT be more like 19. I think the small roll of hardware cloth I bought from the store was a thinner wire than the stuff my hubby snagged from a job site (probably closer to the 23 gauge you're looking at - BTW, that seems like an AMAZING PRICE! Where ever it is you're shopping, stay with that place! WOW!). So the big roll we used on the run is slightly thicker wire. The big difference though between the two (hex vs. hardware cloth) is 1. size of openings 2. hardware cloth (and welded wire) are WELDED where chicken wire is twisted. The welded stuff will always be stronger.
I think the hardware cloth should be sufficient for the bottom. It seems most folks around these boards use it! I don't think that most would purchase the thicker wire, because the cost is outrageous. Again, ours cost $0, so it was a no brainer. The smaller opening compared to the "welded wire" is a bonus to keeping more critters out. Plus it's easier to work with, more bendy LOL. My husband used a hand held grinding wheel to cut through it VERY quickly, although you could use some decent tin snips as well (I had a small electrical wire cutter and cut through the store bought small roll when making my brooder, it took a LONG time to get through one little cut at a time!). That said, if you COULD get the welded wire cheaper it might last longer. But, I think we're talking like a 5 year difference in how long it'll last (ie 10yrs vs. 15yrs as a guestimate). Get the hardware cloth. ;)
Since we JUST did this, I'll tell ya one thing... we used poultry staples to attach it... we wish we'd have just grabbed screws and washers or something of that nature as it would have been MUCH quicker and easier to attach. Poultry staples are a pain in the butt and my hammering hand is still sore.
If the deer fencing you have is as difficult to cut through as you say, I think you'll be FINE with whatever you can buy the cheapest that's metal to make it more difficult for any predators to get through.