If we want a scenario price where we assume a different spot FX, then the ingredients for pricing the new scenario option will be: new spot FX, time to expiry (as original), strike (as original), domestic foreign interest rate (as original), foreign interest rate (as original), and volatility extracted from smile using all these five parameters.
Volatility will clearly be different from the volatility extracted in the original case as in the extraction process we are using a different spot FX rate.
Full Delta is computed by finite differencing using scenario price for spot FX rate shifted up and scenario price for spot FX rate shifted down (the shift in this case is infinitesimal). Even though methodologically different, these two deltas should be relatively close..
Let be the spot FX rate of a currency pair FOREIGN/DOMESTIC (FOR/DOM, f/d) at time t. In the FOR/DOM notation scheme, when one considers, for example, the EUR/USD-spot trading, say, at 1.22 USD per 1 EUR, the currency USD is used to measure 1 EUR, therefore we say that EUR is the underlying (the asset, the foreign currency, the base currency) and USD is the numeraire (the domestic currency, the quote currency) used to value the asset 1 EUR.
Let denote the domestic interest rate, the foreign interest rate, and σ d r f r the volatility parameter. The following continuous geometric Brownian process (under risk-neutral measure) usually models the spot FX rate (here is the standard Brownian motion).
References: