VAR is a conditional mean model and as such has nothing to do with volatility spillovers – unless you apply it on realized variances instead of raw data. In that case it is a model that directly allows for volatility spillovers; see Diebold & Yilmaz "Better to give than to receive: Predictive directional measurement of volatility spillovers" (2012).
Unfortunately, full BEKK-GARCH appears to be a nonsense model; see McAleer "What They Did Not Tell You about Algebraic (Non-) Existence, Mathematical (IR-)Regularity and (Non-) Asymptotic Properties of the Full BEKK Dynamic Conditional Covariance Model" (2019). Diagonal BEKK-GARCH is fine, though (ibid.), but it does not seem to allow for volatility spillovers; see e.g. equations (6), (9) and (11) on p. 5 of Erten et al. "Volatility Spillovers in Emerging Markets During the Global Financial Crisis: Diagonal BEKK Approach" (2012) – even though people use it for the purpose (e.g. Zolfaghari et al. "Volatility spillovers for energy prices: A diagonal BEKK approach" (2020)).
Some people use DCC-GARCH for the purpose, but that model does not allow for volatility spillovers by design (just like the diagonal BEKK-GARCH), as I have explained elsewhere. Moreover, it is also full of problems: McAleer "What They Did Not Tell You about Algebraic (Non-) Existence, Mathematical (IR-)Regularity, and (Non-) Asymptotic Properties of the Dynamic Conditional Correlation (DCC) Model" (2019).
Other models (e.g. GO-GARCH)
There are other multivariate GARCH models out there, as reviewed in Bauwens et al. "Multivariate GARCH models: a survey" (2006) and Silvennoinen & Teräsvirta "Multivariate GARCH models" (2009). Perhaps some of them might work. E.g. consider GO-GARCH; there are at least a couple of packages for estimating the model in R, namely,
rmgarch. I am just afraid McAleer comes out soon and explains how GO-GARCH is full of problems, too :)