Unpaid Invoice To A Business & Debt Collection

I've searched the sub and can find posts similar but not close enough that I feel confident i fully understand, so apologies in advance if this has already been asked and answered a million times!

Tl;dr is that I allegedly owe $1000 to a business that I am not 100% confident I actually owe, and therefore do not want to pay. I believe they may be mistakenly double-billing me for this particular line item, but I can't put together concrete evidence of this because of the way their invoices were presented to me. I'm not sure if my best course of action is to offer to settle with them at a reduced amount just to make this go away, stop engaging with them and let them send it to collections and then try to settle it there, or take some steps on my own to make them stop trying to collect a debt that is only comically only about 0.5% of the total amount I've actually already paid them.

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I had nearly $200k of necessary home renovation completed over 6 months ago, and the contractor now claims that I still owe a balance of $1,000. For multiple reasons, I don't know if I actually owe this and the most likely thing I can come up with is that it may be a bookkeeping error on their end. I paid the rest of the project in full and on time; they signed a waiver of lien that aligns with the amount I've already paid them, and we are well past the timeframe where they could successfully pursue a mechanic's lien.

They are jumping through hoops now to prove I owe this, and I am jumping through hoops to try to figure out how I could have lost track of $1000 in my own bookkeeping if they are correct. I'm not trying to dodge an invoice if I really owe the money, but this is getting ridiculous. $1k on a $200k project is negligible for a large business, and it seems crazy that they haven't just dropped it. It has been mostly amicable communication, but I've stated numerous times I do not want to pay until I'm certain it is actually owed. I finally asked them if they would just drop it so we can all stop wasting our time on this, and they are dodging answering that question.

I have also asked them what their next step in debt collection is, should I decide not to pay. They said it may go to professional collections or legal counsel. I'm guessing legal counsel is an empty threat from a cost-benefit perspective, considering the debt is only $1k.

From my perspective, I see a few options:

1. I attempt to settle with them directly for a partial payment. I hate this SO MUCH because I feel like it is likely I don't actually owe them anything, but I just want them to leave me alone at this point.

2. They deny my offer to pay partially and/or I simply refuse to pay them and it goes to collections. From the debt collector's perspective it's a valid and "proven" debt: there's an invoice with my name and address on it and I didn't pay it. I'd then have to attempt to settle it with the debt collector. If I go this route and pay promptly once they contact me, will it impact my credit score?

3. Is there anything I'm overlooking as far as a way to simply *not* pay anything AND not have it impact my credit? I'm not made of money and $1000 is a lot to throw away after what was already such an expensive and draining ordeal (this wasn't a fun project; it was a necessary one for health and safety, and it was life-altering and will take me yeeeeears to recover financially from this!!!!!!). Could I mail them a copy of their signed lien waiver and proof that I paid in full on all previous invoices that tie to the lien waiver (on their own letterhead), along with a cease and desist letter telling them to stop contacting me about this?

4. Or should I just take the L, pay the $1000, and then go cry about it?

I want to emphasize, I'm really not just trying to get away with not paying $1000 if I owe it. Simply to save myself the headache of dealing with this any further, I'd be fine paying maybe $250-500 to make this go away, but even that annoys me if I don't think I owe them money at all. I just want to know what my rights are and the best course of action to resolve this.

Open to all thoughts and opinions on this, though I would prefer not to get into the nitty gritty details about what is being billed, how i funded this, etc. unless they are necessary because it's just SO MUCH to explain and this post is already long af. Just please trust that I've done my due diligence to try to determine if I did mess up on my end before I started digging my heels in on not wanting to pay.