Making an offer without an agent does not strengthen your position. The listing agreement between the seller and the listing agent is already written...already agreed to. What you might want, what you might propose, will have absolutely no effect whatsoever on that agreement.
What do you have to negotiate with? First, any negotiating skills you may have? Second, hopefully, a knowledge of the comps. I'm not sure how you plan on getting them without an agent. You can try Zillow, but I--and a lot of others here--will caution you that Zillow isn't particularly accurate.
As for using the seller's agent to sell your property: That's fine if you think the seller's agent is competent, knowledgeable, and professional. And if there's a good relationship between the two of you. And perhaps the seller's agent would reduce his/her commission if you were involved in two transactions. But that's in the future. There's nothing you can do about whatever commission the agent and the seller have agreed to on this transaction.
You're asking for "any advice on how to negotiate all this," but it sounds as if you've already decided on your negotiating strategy: Go in without anyone representing you and assume that this will somehow appeal to the seller's agent. That's not the course I'd recommend.
One quick tip: Remember--the seller's agent is only an agent representing the seller. If you're trying to negotiate the price of the house down, whether or not you're using your own agent, the person you must negotiate with and appeal to is the seller. Find out what the seller's needs and motivations are. Put yourself in the seller's shoes. Offer a solution that works not only for you, but also the seller. It could be price...it could be terms...it probably is both. But in any case, shift your negotiating focus from the agent to the seller.