If you want full control for the checkout, Magento is probably going to be the best option. You can also do it with woocommerce. The woo checkout has gotten better and you can tweek it to improve conversions, but you will get more flexibility (if you have a developer) with Magento.

Saying which system has the best conversion rates between those two, BigCommerce and Shopify is hard to say because there are too many factors. To get the best conversions in your cart, the tips would be to ensure you QA the full experience thoroughly on every browser and every device and then iterate consistently to find ways to push more people to complete the sale. You can also use Qualaroo to find out why people do not convert.

Considering the fact that you have a single product, I would have recommended Magento but it's way too much for what you need. WooCommerce is a great option for that.

I currently work with Shopify and Bigcommerce. Both have come a long way and have a great selection of apps that allow you to do all sorts of things. Depending on which plan you're on, Shopify does allow you to change a lot of code within your site.

Hope this help.

LemonStand is the only cloud based eCommerce provider that gives you control over the checkout. You're able to customize the design, steps, functionality and even A/B test the checkout using software like Optimizely.

You can also customize every other aspect of the storefront, for a totally branded shopping experience that's tailored to your audience.