Hi Derek, with the sizes you mention I would pop over to Cleantalk and post the same question. There are a couple of guys on there who have probably forgotten more than the rest of us will ever know; Trevor Ives, Ken Wainwright and John Bolton come to mind straight away and are always quick to give good advice.
You should ideally find out manufacturer and type. As you know tiles aren't always synthetic, some aren't supposed to be wet cleaned at all, so if you can get this info then all the better.
I would look to LM but on an area of this size it would make sense to have at least two operators / machines and maybe one more to help with the pre-spraying / pad rinsing. Oh, as for what to use if LM then Fusion Clean is great.
As for the kitchen, if badly soiled then use a rotary and blue poly brush then wetvac the slurry. Altro offer their own cleaning solutions which work fine, but just stick withing the recommended guidelines and you'll be fine.