I have a member about to hop and build facilities, whats the easiest way to enable them to build facilities from another faction without having to approve every time he starts a build?

Make-over the cities to him.

or ask him to submit the building permit by city design

City design, isn't that still one by one if it's one per city?

Making over's not an option

If its one per city and you really do not feel that making over is an option, you are stuck with increasing the number of people who can accept build requests in order to expedite the acceptance process. There aren't many other options here.

What if he's the cities manager?

The city manager can see the city in the personal inventory, can click on the planet square and view the city from there, can have access to the planet's stats where the city is located.
These are tasks related to a Manager. Building is done by minions ;-) and they must make efforts.

The problem is that the city slab is owned by one party, and the type of building is "owned" by a different party. To avoid hostile take overs, there are no privs that you can give to bypass the build permission step.

It has been suggested several times that managing a city allows you to accept build requests but it has never been added unfortunately.