A good fit is a self-hosted gallery system rather than a full ecommerce proofing service. Based on the suggestions here, Piwigo is the closest match: it’s free, open source, installs on your own website, and with the User Collections plugin clients can select images and you can view their picks in the gallery.

If you later want built-in payment/order handling, a service like Instaproofs is another option, but it’s geared more toward client ordering and takes a fee, so it may be more than you need if you only want image selection.

So for your stated needs—use your own domain, no purchasing required, simple proof selection—a self-hosted Piwigo setup with a selection/collection plugin is likely the best match. It keeps the gallery under your control and avoids the limitations of emailing ZIP files.