I think that you might not be understanding how the spec priority setting is working.
The spec priority is set once for your character. Once you have set it, you don’t need to change it. For example, on my Paladin, my desired priority is Protection, Retribution, Holy. So I set that order either from step 1 of best-in-bags, or from the settings window opened via the filter/funnel icon at the top-middle of the main toolbar at the top of the page.
So no matter which spec I have active, it will always give Protection first priority on stuff, Retribution second, then Holy last.
The gear shown on the best-in-bags result always goes with whatever your active spec is. You can change that by clicking on your character name at the top-left of the page in the big main toolbar – your other specs appear below your name, just click on one to change it.
When you import from our in-game addon by pasting data either into the area on best-in-bags or the main character loader opened by clicking on your name, it will always set your active spec to whatever it is currently in-game.