Thoughts:
We don't want to involve our servants in these dwarf meetings at all. They could very well be a major enemy to their civ, and even revealing we are their masters could sour relations permanently.
We could avoid squabbling in front of the council if we all agreed to only allow one of us to speak to them- if someone talks out of turn he gets put down on the spot by all the heads. If there's a problem head, he won't last more than the first few meetings. Inform the dwarves of this beforehand, and they'll expect it & be prepared as well.
Getting involved in politics is a harder nut to crack. We have the option of being a separate entity, easily manipulated and with little influence over the dwarven society, or we have the option of being directly involved in the danger & intrigue, closer to the manipulation but with a say in it & the society.
I'd say go for a seat on the council.
Back on lizolds: that said, we'll have to reveal our lizards & our position amongst them before we can hope to utilize them within this dwarven civilization. I'd say: if it really seems like a good idea to do so, (hehe, yay vague), once we hear about their thievery problems from them, announce you've made the connection and will stop the raids at once. Wait until after this point before letting them even glimpse a lizold in our presence.
So, what do we want, what can we offer? Ask for the seat on the council, non voting at first of course, and a place in their hall. We might have to gradually work on this. All we can really offer is our combat prowess. Perhaps solving thievery problems, (though that's a risky gambit).
Ask for lodging in their fortress, food & company. Offer our services in defending their, (no, our), home. Simple as that. (To avoid becoming a pet, make sure you get that company. Always keep our dignity paramount, and being civilized works as well.) Maintain we want to be treated as a member of their civ.
yes please to both armor & training!