First, please make sure that when you resimulate sample paths, you are keeping your underlying random samples constant, as in this answer.
For your delta, vega and rho there is some ambiguity in the definition of the greeks. Consider the simple case of delta in the presence of a skew $\sigma(K/S)$, and say that the underlying price right now is $S_0$. We might use
$$
\Delta = \lim_{dS \rightarrow 0} \frac{V(S_0+dS, \sigma(K/S_0))-V(S_0, \sigma(K/S_0))}{dS}
$$
or we might use what is sometimes called the "total delta"
$$
\Delta^{\textrm{tot}} = \lim_{dS \rightarrow 0} \frac{V\left(S_0+dS, \sigma\left(\frac{K}{S_0+dS}\right)\right)-V\left(S_0, \sigma\left(\frac{K}{S_0}\right)\right)}{dS}
$$
In the case of rho and vega, as you note you have similar ambiguity about what model inputs should be changing infinitesimally as part of their definition.
Basically, in the presence of variations from the constant-parameter assumptions of the BSM model, the definition of the greeks is up to you.
What you want to think about is why you are calculating these greeks. The main use of greeks is to help us in hedging. Knowing what the delta is, for example, tells us how much underlying to trade (at a given moment) to lay off some of the hedged position risk.
The role of the greeks in this case is to provide some numbers that can quickly translate to equivalent quantities of liquid securities: underlying for delta, straddles for vega and bonds for rho. The main thing you need to be sure of is that your calculation of equivalents is consistent for the hedging instruments. For example, you would want to calculate your straddle vega using the same definition you used for your basket option vega.
The other role of greeks is to give a sense of portfolio risk to managers who are familiar with thinking of options portfolio risk in these terms. In that case, you want to keep it close to the BSM model, which generally entails
- Not using total delta
- Define vega with a parallel shock to the input implied vol surface
- Define rho with a parallel spot curve shock
