I don't know the full answer, so these are just some thinking-out-loud ideas on how to approach it. I am relying partly on ch9, "Back to basics: an update on the discrete dividend problem", in my book "Option valuation under stochastic volatility II".

First, suppose you knew the market's assumption for the dividend estimates, ex-dates, and interest rates over a particular option period. Then, the Vellekoop-Nieuwenhuis (VN) tree algorithm (discussed in my book chapter) is a fast, convergent GBM option value,

*consistent with discrete dividends*. I haven't tried this, but suspect you can easily invert it for the American-style implied vol parameter for say the option midquote -- just like you can invert the CRR model.

So, then my idea would be to try various methods for the cost-of-carry estimates, and convince yourself that the VN implied vols are not too sensitive to the method. I discuss a couple of cost-of-carry methods in my recent ERP article

here. One is based upon the VIX white paper, and the other is based on put-call parity for general strikes. While put-call parity technically fails under early exercise, I suspect if you stick to close-to-the money options where early exercise is not too likely, you could get some sensible results. A third method would be to simply project the ex-dates and dividends from history, and use the at-the-money option data to infer interest rates. Since the interest rates should be similar across non-dividend paying, non-hard-to-borrow stocks, that should provides some checks.

Then, armed with these various cost-of-carry sets, see how much difference it makes to the VN implied vol inversions.

For arbitrage-free smoothing the IV surface, there is a pretty big literature for this for euro-style options. Now, given a set of American-style implied vols (from the above construction), a common approach in a lot of academic literature is to simply convert all the option prices to euro-style options prices by using the same (American-style) implied vols. Whether this procedure is any good is something to investigate. Once you take this route, then you can apply the arb-free smoothing literature, and then move back to American-style again after smoothing.