What is the industry standard pricing model for CME-traded Eurodollar future (American) options?

To add to @DM63's answer, as a secondary characterisitc, vol may also matter in deciding the european approximation impact. As vol goes to 0, you want to exercise as soon as possible, because the underlying future rate becomes a constant (a martingale with no vol). As you'll receive the same payoff at at date, better to get it earlier (if rates are.positive).

Having traded these options for a number of years I have some insight. It’s my belief that those that make a living specifically out of these options do have tree-style models that take into account early exercise. On the other hand , those that have occasional use of these options (such as interest rate derivatives dealers who might use them to hedge otc derivatives) mostly assume that the options are European, because implementing the full American model is cumbersome within a multi- product portfolio.

It is possible to show that the error committed in assuming the option is European is not very large. Consider the case of a call option on the Mar2020 contract (approx 1year to expiration) , struck at 9450, with the contract trading at 9750. This contract is deep in the money (actually, 300bp In the money). Left unexercised, it is worth 300bp, discounted for one year. However , upon exercise , you get delivered the underlying futures contract, together with 300bp of immediate variation margin. This situation is worth 300bp , not discounted. Hence , it is my experience that the modeling error only shows up for long dated, deep in the money options , and then only when interest rates are high enough to make a discounting difference.

I hope that helps.