I would not let the chosen model depend on the payoff function.
For instance, consider a financial derivative where the underlying asset is a perfectly deterministic function of time. Then, your payoff function is also deterministic. So why model this underlying asset with a stochastic process?
The preferred model is often a trade-off between accuracy/market fit and complexity. Therefore, I would start with a basic model and try to fit it to market quotes. If no quotes are available, try to use similar products for which the quotes are available and fit those. When the fit does not suffice your needs, try adding little bits of complexity to increase the fit of your model to the market data.
I think in your case, start with a plain diffusion process to check if you can implement a working model. In this case, you can even derive a closed-form solution, I think. This will probably result in a poor volatility fit but gives you a good starting point to improve your model and find out which approach will work.
Then maybe increase to a jump-diffusion model, and afterward include stochastic volatility.
Especially, if your underlying is possibly dependent on the market interest rates, it also makes sense to think about incorporating stochastic interest rates.
Note that your model needs to be fitted to obtain workable results. Therefore, it makes no sense to include much complexity if the model cannot be fitted.
I hope you find it useful.