Can someone point me to a white-label website builder which allows custom payment gateway integration?

Apart from what you want, when choosing a white label website builder, cost will be a crucial deciding factor. Find a reseller program that matches your cost expectations but do not settle for anything less in terms of quality. As a designer, you will probably have adopted a particular style for creating websites. Check out the site templates offered under Strikingly to see if our designs match your own. In this way it would be easier for you to use our templates and customize them to reflect your unique brand.
Customer support is also especially important. You must be provided with a good number of support options in case you or your client runs into an issue during website development and management. Strikingly offers 24/7 support for all accounts along with a highly informative knowledge base so you will always have answers when you need them.
Check out these following links which I believe will be of great help to you:
1. https://agency.weblium.com/
2. https://sitegalore.com/
3. https://www.duda.co/website-builder/white-label
4. https://webwave.me/
5. https://www.ucraft.com/whitelabel
6. https://offsprout.com/
7. https://www.motocms.com/customer-services/reseller-white-label-website-builder/
8. https://www.monosolutions.com/platform/website-builder

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Hi,

You can start with WordPress or another CMS if you want to build the 3rd party payment integration in PHP. You can use a WaaS to create the white label website builder and add your own PG via WooCommerce.

https://www.ecwid.com/partners/white-label has a white label service and has a lot of integrations, or perhaps this: https://simvoly.com/whitelabel-website-builder

Though to be fair the first option has hundreds of payment plugins though Woo, so unless it's your own crypto, you might find there is one already available for your payment gateway.
https://woocommerce.com/product-category/woocommerce-extensions/payment-gateways/

If you want any further information or help on your project, please get in contact and I'd be happy to try and find a solution with you.

Today, there are even more options for white-label ecommerce platforms. One standout option is fesh.store, which offers a unique pricing model. Instead of paying for each store individually, you pay a small monthly fee that includes a pre-determined number of stores. This allows you to maintain predictable costs with a fixed price, making it easier to budget as you scale your business.

White label program page: https://fesh.store/white-label-ecommerce-platform

The key here is to find the optimal partnership model - whether it's per-store payment or a fixed monthly fee - and to evaluate the system features provided.
You can't just offer a store anymore; you need to provide solutions for SEO and marketing, as these are the main challenges for store owners today.