If this happens, it will happen with the President also being a Senator. In fact, arguably that would fit the original 1789 design better, where the Senate was understood to be very different from the House. The House was thought of as a representative, domestic, purely legislative body, whereas the Senate was supposed to be consultative, elite, and intimately involved along with the President in the direction of foreign affairs.
The President probably largely appoints his cabinet from the Senate, the cabinet continuing to sit as Senators.
The position of the Vice President is going to be even more anomalous, given that he will now preside over the President in the Senate. Maybe you resolve this by having a Washington establish the precedent that the President would attend the Senate from time to time but refrains from voting or debating. Expect a TR figure to controversially end the precedent. At which point you maybe get an amendment that the VP is appointed by the President in consultation with the Senate.