PalacioHill

use the closed-form formula for european up-and-in call. in theory, it should give you a lower bound. in further theory, once the call gets knocked in and becomes an ordinary american call, i dont see why you can't apply the same no arbitrage principle - i.e. american call on a non-dividend paying stock won't be exercised prior to maturity. so if the underlying doesn't pay dividends, then the closed-form european up-and-in price should be identical to the american up-and-in price.

in practice, the price is even lower (as opposed to higher) - you work for a private company (the underlying is illiquid), and this is executive comp (so the option itself is likely illiquid or with some transfer restrictions). there is no point in hunting down complex lattice models - the lack of liquidity in your case makes this exercise pointless. in summary, just use the closed-form european form.