well supply line "tee"

I purchased a property with a pole barn that was under construction. The existing well/pump use to feed a double wide that no longer present. This end of the pipe was capped with a faucet. It appears the prior owner ran a new under ground pipe from the well to the pole barn by teeing into the existing underground line. This new line was to feed a Bladder tank located in the pole barn to supply water to hot water heater and bathroom.

1) Will this system work properly with a check valve at the bladder tank. It would think the check valve would prevent a pressure drop at the switch if the out side valve was turned on.

2) should I dig up the Tee and eliminate the exterior faucet and re-plumb after the bladder tank?

You don't need a check valve at the bladder tank. If the pump doesn't already have a check valve, you will need to install one at the pump, which is the only one you need. Then because there is no check valve at the tank, the faucet will work fine where it is.