Right-click a shulker box in your inventory and its contents open on a chest screen. Take things out, put things in, close it. The box never leaves your pack.
The same idea Bundle Browser has for bundles, for the container that actually gets carried around full.
Only with an empty hand, and only the right button. What that costs is picking a box up with the right button - which the left button already does, and which nobody does on purpose: a shulker box does not stack, so the half-a-stack that right-click exists for has never applied to one.
Bundle Browser is client-side and works on any server. This is server-side and works for any client, including one that has never heard of it.
That is not a preference, it is the only way round that exists. Vanilla lets you click items out of a bundle, so a client mod can drive that with ordinary clicks; nothing in vanilla takes an item out of a shulker box that has not been placed, so there is no interaction for a client to drive and the server has to be the one holding the box open.
What the player gets is a real container menu, not a drawn one: the chest screen they already know, with the shulker's own items in it, and every click is one vanilla already knows how to send.