Once we've initiated the process of paying with coupons (or via 1099A), should we stop sending additional payments? If yes, is there a known way to receive those payments back once they acknowledge our coupons and/or 1099A?
That would depend on you. There may be short term negative consequences of stopping payment such as: negative credit reporting, potentially even losing your property if the debt is secured by a car or house for example.
Typically one has to enforce their administrative process before they stop and zero out your account. Until enforced, if I want to advert the negative effect of late payments, I would pay under fraud, duress and extortion. (If paying by check, I would write on it "under fraud, duress and extortion" in the memo field)
In order to move towards enforcement, one would have to established a new contract and show failure to perform under that contract... that is why the next step is a conditional acceptance, followed by a Treasury Contract and lien security against the accused etc.. (Courses on these topics coming soon)