Convertible Bond in Foreign Currency - Quanto Adjustment

I need to value the following convertible bond:

The bond notional and interest is denoted in USD, but is convertible into Euro denominated equity.

Normally, I would value such a bond with a binomial tree as it takes the form of an American option, however I do not know what sort of adjustments I can make to take account of the fact that the strike is in a foreign currency.

My gut feeling says to use the forward fx rates available and then make some sort of quanto adjustment to the volatility going into the model, however I am unsure.

If the USD denominated bond is simply convertible into a “Euro denominated equity” then it is not quanto but composite: upon conversion the bond holder gets $x$ units of equity, where $x$ is the conversion ratio, each unit being worth in USD the composite price $S_{\text{USD}}(t) = S_{\text{EUR}}(t) \times \text{eurusd}(t)$, so you are only left with building a model (e.g. a binomial tree) for $S_{\text{USD}}(t)$ under the USD risk neutral measure.
If by any chance the equity is quoted in the US in the form of an ADR and there are quoted derivatives on the ADR then you will have access to forward and implied volatility directly for $S_{\text{USD}}(t)$.
Otherwise you will need to start from forwards and volatility for $S_{\text{EUR}}(t)$, convert to USD forwards by multiplying by the forward FX, and compute the USD equity price volatility from the formula $\sigma_{\text{USD}} = \sqrt{\sigma_{\text{EUR}}^2 + \sigma_{\text{eurusd}}^2 + 2 \rho \sigma_{\text{EUR}} \sigma_{\text{eurusd}}}$ where $\sigma_{\text{EUR}}$ is the EUR equity price volatility, $\sigma_{\text{eurusd}}$ is the FX volatility, and $\rho$ is the correlation between the EUR equity price and the FX. Uncertainty on estimating the correlation will be the main problem since it will likely have to be done historically.
Note that if there are EUR denominated features in the convertible, such as early redemption thresholds based on the EUR equity price, then the problem does become quanto and also requires joint modeling of $S_{\text{EUR}}(t)$ and $\text{eurusd}(t)$ under the USD risk neutral measure, with a quanto adjustment for the dynamics of $S_{\text{EUR}}(t)$.